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Theorem hashf1lem1 13809

Description: Lemma for hashf1 13811. (Contributed by Mario Carneiro, 17-Apr-2015.)

**Hypotheses**
Ref	Expression
hashf1lem2.1	⊢ (𝜑 → 𝐴 ∈ Fin)
hashf1lem2.2	⊢ (𝜑 → 𝐵 ∈ Fin)
hashf1lem2.3	⊢ (𝜑 → ¬ 𝑧 ∈ 𝐴)
hashf1lem2.4	⊢ (𝜑 → ((♯‘𝐴) + 1) ≤ (♯‘𝐵))
hashf1lem1.5	⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵)

**Assertion**
Ref	Expression
hashf1lem1	⊢ (𝜑 → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ≈ (𝐵 ∖ ran 𝐹))

Distinct variable groups: 𝑧,𝑓 𝐴,𝑓 𝐵,𝑓 𝜑,𝑓 𝑓,𝐹 Allowed substitution hints: 𝜑(𝑧) 𝐴(𝑧) 𝐵(𝑧) 𝐹(𝑧)

Proof of Theorem hashf1lem1 Dummy variables 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1f 6549	. . . . . 6 ⊢ (𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵 → 𝑓:(𝐴 ∪ {𝑧})⟶𝐵)
2	1	adantl 485	. . . . 5 ⊢ (((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) → 𝑓:(𝐴 ∪ {𝑧})⟶𝐵)
3		hashf1lem2.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ Fin)
4		hashf1lem2.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ Fin)
5		snfi 8577	. . . . . . 7 ⊢ {𝑧} ∈ Fin
6		unfi 8769	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝐴 ∪ {𝑧}) ∈ Fin)
7	4, 5, 6	sylancl 589	. . . . . 6 ⊢ (𝜑 → (𝐴 ∪ {𝑧}) ∈ Fin)
8	3, 7	elmapd 8403	. . . . 5 ⊢ (𝜑 → (𝑓 ∈ (𝐵 ↑_m (𝐴 ∪ {𝑧})) ↔ 𝑓:(𝐴 ∪ {𝑧})⟶𝐵))
9	2, 8	syl5ibr 249	. . . 4 ⊢ (𝜑 → (((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) → 𝑓 ∈ (𝐵 ↑_m (𝐴 ∪ {𝑧}))))
10	9	abssdv 3996	. . 3 ⊢ (𝜑 → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ⊆ (𝐵 ↑_m (𝐴 ∪ {𝑧})))
11		ovex 7168	. . 3 ⊢ (𝐵 ↑_m (𝐴 ∪ {𝑧})) ∈ V
12		ssexg 5191	. . 3 ⊢ (({𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ⊆ (𝐵 ↑_m (𝐴 ∪ {𝑧})) ∧ (𝐵 ↑_m (𝐴 ∪ {𝑧})) ∈ V) → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∈ V)
13	10, 11, 12	sylancl 589	. 2 ⊢ (𝜑 → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∈ V)
14		difexg 5195	. . 3 ⊢ (𝐵 ∈ Fin → (𝐵 ∖ ran 𝐹) ∈ V)
15	3, 14	syl 17	. 2 ⊢ (𝜑 → (𝐵 ∖ ran 𝐹) ∈ V)
16		vex 3444	. . . 4 ⊢ 𝑔 ∈ V
17		reseq1 5812	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑓 ↾ 𝐴) = (𝑔 ↾ 𝐴))
18	17	eqeq1d 2800	. . . . 5 ⊢ (𝑓 = 𝑔 → ((𝑓 ↾ 𝐴) = 𝐹 ↔ (𝑔 ↾ 𝐴) = 𝐹))
19		f1eq1 6544	. . . . 5 ⊢ (𝑓 = 𝑔 → (𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵 ↔ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵))
20	18, 19	anbi12d 633	. . . 4 ⊢ (𝑓 = 𝑔 → (((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ↔ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)))
21	16, 20	elab 3615	. . 3 ⊢ (𝑔 ∈ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ↔ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵))
22		f1f 6549	. . . . . . 7 ⊢ (𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵 → 𝑔:(𝐴 ∪ {𝑧})⟶𝐵)
23	22	ad2antll 728	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → 𝑔:(𝐴 ∪ {𝑧})⟶𝐵)
24		ssun2 4100	. . . . . . 7 ⊢ {𝑧} ⊆ (𝐴 ∪ {𝑧})
25		vex 3444	. . . . . . . 8 ⊢ 𝑧 ∈ V
26	25	snss 4679	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 ∪ {𝑧}) ↔ {𝑧} ⊆ (𝐴 ∪ {𝑧}))
27	24, 26	mpbir 234	. . . . . 6 ⊢ 𝑧 ∈ (𝐴 ∪ {𝑧})
28		ffvelrn 6826	. . . . . 6 ⊢ ((𝑔:(𝐴 ∪ {𝑧})⟶𝐵 ∧ 𝑧 ∈ (𝐴 ∪ {𝑧})) → (𝑔‘𝑧) ∈ 𝐵)
29	23, 27, 28	sylancl 589	. . . . 5 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → (𝑔‘𝑧) ∈ 𝐵)
30		hashf1lem2.3	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝐴)
31	30	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → ¬ 𝑧 ∈ 𝐴)
32		df-ima 5532	. . . . . . . . 9 ⊢ (𝑔 “ 𝐴) = ran (𝑔 ↾ 𝐴)
33		simprl 770	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → (𝑔 ↾ 𝐴) = 𝐹)
34	33	rneqd 5772	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → ran (𝑔 ↾ 𝐴) = ran 𝐹)
35	32, 34	syl5eq 2845	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → (𝑔 “ 𝐴) = ran 𝐹)
36	35	eleq2d 2875	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → ((𝑔‘𝑧) ∈ (𝑔 “ 𝐴) ↔ (𝑔‘𝑧) ∈ ran 𝐹))
37		simprr 772	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)
38	27	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → 𝑧 ∈ (𝐴 ∪ {𝑧}))
39		ssun1 4099	. . . . . . . . 9 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝑧})
40	39	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → 𝐴 ⊆ (𝐴 ∪ {𝑧}))
41		f1elima 6999	. . . . . . . 8 ⊢ ((𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵 ∧ 𝑧 ∈ (𝐴 ∪ {𝑧}) ∧ 𝐴 ⊆ (𝐴 ∪ {𝑧})) → ((𝑔‘𝑧) ∈ (𝑔 “ 𝐴) ↔ 𝑧 ∈ 𝐴))
42	37, 38, 40, 41	syl3anc 1368	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → ((𝑔‘𝑧) ∈ (𝑔 “ 𝐴) ↔ 𝑧 ∈ 𝐴))
43	36, 42	bitr3d 284	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → ((𝑔‘𝑧) ∈ ran 𝐹 ↔ 𝑧 ∈ 𝐴))
44	31, 43	mtbird 328	. . . . 5 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → ¬ (𝑔‘𝑧) ∈ ran 𝐹)
45	29, 44	eldifd 3892	. . . 4 ⊢ ((𝜑 ∧ ((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)) → (𝑔‘𝑧) ∈ (𝐵 ∖ ran 𝐹))
46	45	ex 416	. . 3 ⊢ (𝜑 → (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) → (𝑔‘𝑧) ∈ (𝐵 ∖ ran 𝐹)))
47	21, 46	syl5bi 245	. 2 ⊢ (𝜑 → (𝑔 ∈ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} → (𝑔‘𝑧) ∈ (𝐵 ∖ ran 𝐹)))
48		hashf1lem1.5	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵)
49		f1f 6549	. . . . . . 7 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
50	48, 49	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
51	50	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → 𝐹:𝐴⟶𝐵)
52		vex 3444	. . . . . . . 8 ⊢ 𝑥 ∈ V
53	25, 52	f1osn 6629	. . . . . . 7 ⊢ {⟨𝑧, 𝑥⟩}:{𝑧}–1-1-onto→{𝑥}
54		f1of 6590	. . . . . . 7 ⊢ ({⟨𝑧, 𝑥⟩}:{𝑧}–1-1-onto→{𝑥} → {⟨𝑧, 𝑥⟩}:{𝑧}⟶{𝑥})
55	53, 54	ax-mp 5	. . . . . 6 ⊢ {⟨𝑧, 𝑥⟩}:{𝑧}⟶{𝑥}
56		eldifi 4054	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 ∖ ran 𝐹) → 𝑥 ∈ 𝐵)
57	56	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → 𝑥 ∈ 𝐵)
58	57	snssd 4702	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → {𝑥} ⊆ 𝐵)
59		fss 6501	. . . . . 6 ⊢ (({⟨𝑧, 𝑥⟩}:{𝑧}⟶{𝑥} ∧ {𝑥} ⊆ 𝐵) → {⟨𝑧, 𝑥⟩}:{𝑧}⟶𝐵)
60	55, 58, 59	sylancr 590	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → {⟨𝑧, 𝑥⟩}:{𝑧}⟶𝐵)
61		res0 5822	. . . . . . 7 ⊢ (𝐹 ↾ ∅) = ∅
62		res0 5822	. . . . . . 7 ⊢ ({⟨𝑧, 𝑥⟩} ↾ ∅) = ∅
63	61, 62	eqtr4i 2824	. . . . . 6 ⊢ (𝐹 ↾ ∅) = ({⟨𝑧, 𝑥⟩} ↾ ∅)
64		disjsn 4607	. . . . . . . . 9 ⊢ ((𝐴 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝐴)
65	30, 64	sylibr 237	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∩ {𝑧}) = ∅)
66	65	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐴 ∩ {𝑧}) = ∅)
67	66	reseq2d 5818	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ↾ (𝐴 ∩ {𝑧})) = (𝐹 ↾ ∅))
68	66	reseq2d 5818	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ({⟨𝑧, 𝑥⟩} ↾ (𝐴 ∩ {𝑧})) = ({⟨𝑧, 𝑥⟩} ↾ ∅))
69	63, 67, 68	3eqtr4a 2859	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ↾ (𝐴 ∩ {𝑧})) = ({⟨𝑧, 𝑥⟩} ↾ (𝐴 ∩ {𝑧})))
70		fresaunres1 6525	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ {⟨𝑧, 𝑥⟩}:{𝑧}⟶𝐵 ∧ (𝐹 ↾ (𝐴 ∩ {𝑧})) = ({⟨𝑧, 𝑥⟩} ↾ (𝐴 ∩ {𝑧}))) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹)
71	51, 60, 69, 70	syl3anc 1368	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹)
72		f1f1orn 6601	. . . . . . . . 9 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran 𝐹)
73	48, 72	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→ran 𝐹)
74	73	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → 𝐹:𝐴–1-1-onto→ran 𝐹)
75	53	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → {⟨𝑧, 𝑥⟩}:{𝑧}–1-1-onto→{𝑥})
76		eldifn 4055	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ∖ ran 𝐹) → ¬ 𝑥 ∈ ran 𝐹)
77	76	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ¬ 𝑥 ∈ ran 𝐹)
78		disjsn 4607	. . . . . . . 8 ⊢ ((ran 𝐹 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ ran 𝐹)
79	77, 78	sylibr 237	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (ran 𝐹 ∩ {𝑥}) = ∅)
80		f1oun 6609	. . . . . . 7 ⊢ (((𝐹:𝐴–1-1-onto→ran 𝐹 ∧ {⟨𝑧, 𝑥⟩}:{𝑧}–1-1-onto→{𝑥}) ∧ ((𝐴 ∩ {𝑧}) = ∅ ∧ (ran 𝐹 ∩ {𝑥}) = ∅)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}))
81	74, 75, 66, 79, 80	syl22anc 837	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}))
82		f1of1 6589	. . . . . 6 ⊢ ((𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→(ran 𝐹 ∪ {𝑥}))
83	81, 82	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→(ran 𝐹 ∪ {𝑥}))
84	51	frnd 6494	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ran 𝐹 ⊆ 𝐵)
85	84, 58	unssd 4113	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (ran 𝐹 ∪ {𝑥}) ⊆ 𝐵)
86		f1ss 6555	. . . . 5 ⊢ (((𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→(ran 𝐹 ∪ {𝑥}) ∧ (ran 𝐹 ∪ {𝑥}) ⊆ 𝐵) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→𝐵)
87	83, 85, 86	syl2anc 587	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→𝐵)
88		fex 6966	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ Fin) → 𝐹 ∈ V)
89	50, 4, 88	syl2anc 587	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ V)
90	89	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → 𝐹 ∈ V)
91		snex 5297	. . . . . 6 ⊢ {⟨𝑧, 𝑥⟩} ∈ V
92		unexg 7452	. . . . . 6 ⊢ ((𝐹 ∈ V ∧ {⟨𝑧, 𝑥⟩} ∈ V) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ V)
93	90, 91, 92	sylancl 589	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ V)
94		reseq1 5812	. . . . . . . 8 ⊢ (𝑓 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) → (𝑓 ↾ 𝐴) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴))
95	94	eqeq1d 2800	. . . . . . 7 ⊢ (𝑓 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) → ((𝑓 ↾ 𝐴) = 𝐹 ↔ ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹))
96		f1eq1 6544	. . . . . . 7 ⊢ (𝑓 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) → (𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵 ↔ (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→𝐵))
97	95, 96	anbi12d 633	. . . . . 6 ⊢ (𝑓 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) → (((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵) ↔ (((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹 ∧ (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→𝐵)))
98	97	elabg 3614	. . . . 5 ⊢ ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ V → ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ↔ (((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹 ∧ (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→𝐵)))
99	93, 98	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ↔ (((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹 ∧ (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→𝐵)))
100	71, 87, 99	mpbir2and 712	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)})
101	100	ex 416	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐵 ∖ ran 𝐹) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)}))
102	21	anbi1i 626	. . 3 ⊢ ((𝑔 ∈ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) ↔ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)))
103		simprlr 779	. . . . . . 7 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵)
104		f1fn 6550	. . . . . . 7 ⊢ (𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵 → 𝑔 Fn (𝐴 ∪ {𝑧}))
105	103, 104	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → 𝑔 Fn (𝐴 ∪ {𝑧}))
106	81	adantrl 715	. . . . . . 7 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}))
107		f1ofn 6591	. . . . . . 7 ⊢ ((𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) Fn (𝐴 ∪ {𝑧}))
108	106, 107	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) Fn (𝐴 ∪ {𝑧}))
109		eqfnfv 6779	. . . . . 6 ⊢ ((𝑔 Fn (𝐴 ∪ {𝑧}) ∧ (𝐹 ∪ {⟨𝑧, 𝑥⟩}) Fn (𝐴 ∪ {𝑧})) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ ∀𝑦 ∈ (𝐴 ∪ {𝑧})(𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦)))
110	105, 108, 109	syl2anc 587	. . . . 5 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ ∀𝑦 ∈ (𝐴 ∪ {𝑧})(𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦)))
111		fvres 6664	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐴 → ((𝑔 ↾ 𝐴)‘𝑦) = (𝑔‘𝑦))
112	111	eqcomd 2804	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐴 → (𝑔‘𝑦) = ((𝑔 ↾ 𝐴)‘𝑦))
113		simprll 778	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝑔 ↾ 𝐴) = 𝐹)
114	113	fveq1d 6647	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ((𝑔 ↾ 𝐴)‘𝑦) = (𝐹‘𝑦))
115	112, 114	sylan9eqr 2855	. . . . . . . . 9 ⊢ (((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦 ∈ 𝐴) → (𝑔‘𝑦) = (𝐹‘𝑦))
116	48	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦 ∈ 𝐴) → 𝐹:𝐴–1-1→𝐵)
117		f1fn 6550	. . . . . . . . . . 11 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)
118	116, 117	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦 ∈ 𝐴) → 𝐹 Fn 𝐴)
119	25, 52	fnsn 6382	. . . . . . . . . . 11 ⊢ {⟨𝑧, 𝑥⟩} Fn {𝑧}
120	119	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦 ∈ 𝐴) → {⟨𝑧, 𝑥⟩} Fn {𝑧})
121	65	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦 ∈ 𝐴) → (𝐴 ∩ {𝑧}) = ∅)
122		simpr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴)
123		fvun1 6729	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ {⟨𝑧, 𝑥⟩} Fn {𝑧} ∧ ((𝐴 ∩ {𝑧}) = ∅ ∧ 𝑦 ∈ 𝐴)) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) = (𝐹‘𝑦))
124	118, 120, 121, 122, 123	syl112anc 1371	. . . . . . . . 9 ⊢ (((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦 ∈ 𝐴) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) = (𝐹‘𝑦))
125	115, 124	eqtr4d 2836	. . . . . . . 8 ⊢ (((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦 ∈ 𝐴) → (𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦))
126	125	ralrimiva 3149	. . . . . . 7 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦))
127	126	biantrurd 536	. . . . . 6 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (∀𝑦 ∈ {𝑧} (𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ (∀𝑦 ∈ 𝐴 (𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ∧ ∀𝑦 ∈ {𝑧} (𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦))))
128		ralunb 4118	. . . . . 6 ⊢ (∀𝑦 ∈ (𝐴 ∪ {𝑧})(𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ (∀𝑦 ∈ 𝐴 (𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ∧ ∀𝑦 ∈ {𝑧} (𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦)))
129	127, 128	syl6bbr 292	. . . . 5 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (∀𝑦 ∈ {𝑧} (𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ ∀𝑦 ∈ (𝐴 ∪ {𝑧})(𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦)))
130	50	fdmd 6497	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐹 = 𝐴)
131	130	eleq2d 2875	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ dom 𝐹 ↔ 𝑧 ∈ 𝐴))
132	30, 131	mtbird 328	. . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑧 ∈ dom 𝐹)
133	132	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ¬ 𝑧 ∈ dom 𝐹)
134		fsnunfv 6926	. . . . . . . 8 ⊢ ((𝑧 ∈ V ∧ 𝑥 ∈ V ∧ ¬ 𝑧 ∈ dom 𝐹) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧) = 𝑥)
135	25, 52, 133, 134	mp3an12i 1462	. . . . . . 7 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧) = 𝑥)
136	135	eqeq2d 2809	. . . . . 6 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ((𝑔‘𝑧) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧) ↔ (𝑔‘𝑧) = 𝑥))
137		fveq2 6645	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝑔‘𝑦) = (𝑔‘𝑧))
138		fveq2 6645	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧))
139	137, 138	eqeq12d 2814	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ((𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ (𝑔‘𝑧) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧)))
140	25, 139	ralsn 4579	. . . . . 6 ⊢ (∀𝑦 ∈ {𝑧} (𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ (𝑔‘𝑧) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧))
141		eqcom 2805	. . . . . 6 ⊢ (𝑥 = (𝑔‘𝑧) ↔ (𝑔‘𝑧) = 𝑥)
142	136, 140, 141	3bitr4g 317	. . . . 5 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (∀𝑦 ∈ {𝑧} (𝑔‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ 𝑥 = (𝑔‘𝑧)))
143	110, 129, 142	3bitr2d 310	. . . 4 ⊢ ((𝜑 ∧ (((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ 𝑥 = (𝑔‘𝑧)))
144	143	ex 416	. . 3 ⊢ (𝜑 → ((((𝑔 ↾ 𝐴) = 𝐹 ∧ 𝑔:(𝐴 ∪ {𝑧})–1-1→𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ 𝑥 = (𝑔‘𝑧))))
145	102, 144	syl5bi 245	. 2 ⊢ (𝜑 → ((𝑔 ∈ {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ 𝑥 = (𝑔‘𝑧))))
146	13, 15, 47, 101, 145	en3d 8529	1 ⊢ (𝜑 → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ≈ (𝐵 ∖ ran 𝐹))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {cab 2776 ∀wral 3106 Vcvv 3441 ∖ cdif 3878 ∪ cun 3879 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 {csn 4525 ⟨cop 4531 class class class wbr 5030 dom cdm 5519 ran crn 5520 ↾ cres 5521 “ cima 5522 Fn wfn 6319 ⟶wf 6320 –1-1→wf1 6321 –1-1-onto→wf1o 6323 ‘cfv 6324 (class class class)co 7135 ↑_m cmap 8389 ≈ cen 8489 Fincfn 8492 1c1 10527 + caddc 10529 ≤ cle 10665 ♯chash 13686

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-fin 8496

This theorem is referenced by: hashf1lem2 13810

W3C validator