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Theorem hashf1lem1OLD 13920
Description: Obsolete version of hashf1lem1 13919 as of 7-Aug-2024. (Contributed by Mario Carneiro, 17-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
hashf1lem2.1 (𝜑𝐴 ∈ Fin)
hashf1lem2.2 (𝜑𝐵 ∈ Fin)
hashf1lem2.3 (𝜑 → ¬ 𝑧𝐴)
hashf1lem2.4 (𝜑 → ((♯‘𝐴) + 1) ≤ (♯‘𝐵))
hashf1lem1.5 (𝜑𝐹:𝐴1-1𝐵)
Assertion
Ref Expression
hashf1lem1OLD (𝜑 → {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ≈ (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 hashf1lem1OLD
Dummy variables 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1f 6585 . . . . . 6 (𝑓:(𝐴 ∪ {𝑧})–1-1𝐵𝑓:(𝐴 ∪ {𝑧})⟶𝐵)
21adantl 485 . . . . 5 (((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵) → 𝑓:(𝐴 ∪ {𝑧})⟶𝐵)
3 hashf1lem2.2 . . . . . 6 (𝜑𝐵 ∈ Fin)
4 hashf1lem2.1 . . . . . . 7 (𝜑𝐴 ∈ Fin)
5 snfi 8655 . . . . . . 7 {𝑧} ∈ Fin
6 unfi 8784 . . . . . . 7 ((𝐴 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝐴 ∪ {𝑧}) ∈ Fin)
74, 5, 6sylancl 589 . . . . . 6 (𝜑 → (𝐴 ∪ {𝑧}) ∈ Fin)
83, 7elmapd 8464 . . . . 5 (𝜑 → (𝑓 ∈ (𝐵m (𝐴 ∪ {𝑧})) ↔ 𝑓:(𝐴 ∪ {𝑧})⟶𝐵))
92, 8syl5ibr 249 . . . 4 (𝜑 → (((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵) → 𝑓 ∈ (𝐵m (𝐴 ∪ {𝑧}))))
109abssdv 3968 . . 3 (𝜑 → {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ⊆ (𝐵m (𝐴 ∪ {𝑧})))
11 ovex 7216 . . 3 (𝐵m (𝐴 ∪ {𝑧})) ∈ V
12 ssexg 5201 . . 3 (({𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ⊆ (𝐵m (𝐴 ∪ {𝑧})) ∧ (𝐵m (𝐴 ∪ {𝑧})) ∈ V) → {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ∈ V)
1310, 11, 12sylancl 589 . 2 (𝜑 → {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ∈ V)
14 difexg 5205 . . 3 (𝐵 ∈ Fin → (𝐵 ∖ ran 𝐹) ∈ V)
153, 14syl 17 . 2 (𝜑 → (𝐵 ∖ ran 𝐹) ∈ V)
16 vex 3404 . . . 4 𝑔 ∈ V
17 reseq1 5829 . . . . . 6 (𝑓 = 𝑔 → (𝑓𝐴) = (𝑔𝐴))
1817eqeq1d 2741 . . . . 5 (𝑓 = 𝑔 → ((𝑓𝐴) = 𝐹 ↔ (𝑔𝐴) = 𝐹))
19 f1eq1 6580 . . . . 5 (𝑓 = 𝑔 → (𝑓:(𝐴 ∪ {𝑧})–1-1𝐵𝑔:(𝐴 ∪ {𝑧})–1-1𝐵))
2018, 19anbi12d 634 . . . 4 (𝑓 = 𝑔 → (((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵) ↔ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)))
2116, 20elab 3578 . . 3 (𝑔 ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ↔ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵))
22 f1f 6585 . . . . . . 7 (𝑔:(𝐴 ∪ {𝑧})–1-1𝐵𝑔:(𝐴 ∪ {𝑧})⟶𝐵)
2322ad2antll 729 . . . . . 6 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → 𝑔:(𝐴 ∪ {𝑧})⟶𝐵)
24 ssun2 4073 . . . . . . 7 {𝑧} ⊆ (𝐴 ∪ {𝑧})
25 vex 3404 . . . . . . . 8 𝑧 ∈ V
2625snss 4684 . . . . . . 7 (𝑧 ∈ (𝐴 ∪ {𝑧}) ↔ {𝑧} ⊆ (𝐴 ∪ {𝑧}))
2724, 26mpbir 234 . . . . . 6 𝑧 ∈ (𝐴 ∪ {𝑧})
28 ffvelrn 6872 . . . . . 6 ((𝑔:(𝐴 ∪ {𝑧})⟶𝐵𝑧 ∈ (𝐴 ∪ {𝑧})) → (𝑔𝑧) ∈ 𝐵)
2923, 27, 28sylancl 589 . . . . 5 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → (𝑔𝑧) ∈ 𝐵)
30 hashf1lem2.3 . . . . . . 7 (𝜑 → ¬ 𝑧𝐴)
3130adantr 484 . . . . . 6 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → ¬ 𝑧𝐴)
32 df-ima 5548 . . . . . . . . 9 (𝑔𝐴) = ran (𝑔𝐴)
33 simprl 771 . . . . . . . . . 10 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → (𝑔𝐴) = 𝐹)
3433rneqd 5791 . . . . . . . . 9 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → ran (𝑔𝐴) = ran 𝐹)
3532, 34syl5eq 2786 . . . . . . . 8 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → (𝑔𝐴) = ran 𝐹)
3635eleq2d 2819 . . . . . . 7 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → ((𝑔𝑧) ∈ (𝑔𝐴) ↔ (𝑔𝑧) ∈ ran 𝐹))
37 simprr 773 . . . . . . . 8 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → 𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)
3827a1i 11 . . . . . . . 8 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → 𝑧 ∈ (𝐴 ∪ {𝑧}))
39 ssun1 4072 . . . . . . . . 9 𝐴 ⊆ (𝐴 ∪ {𝑧})
4039a1i 11 . . . . . . . 8 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → 𝐴 ⊆ (𝐴 ∪ {𝑧}))
41 f1elima 7045 . . . . . . . 8 ((𝑔:(𝐴 ∪ {𝑧})–1-1𝐵𝑧 ∈ (𝐴 ∪ {𝑧}) ∧ 𝐴 ⊆ (𝐴 ∪ {𝑧})) → ((𝑔𝑧) ∈ (𝑔𝐴) ↔ 𝑧𝐴))
4237, 38, 40, 41syl3anc 1372 . . . . . . 7 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → ((𝑔𝑧) ∈ (𝑔𝐴) ↔ 𝑧𝐴))
4336, 42bitr3d 284 . . . . . 6 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → ((𝑔𝑧) ∈ ran 𝐹𝑧𝐴))
4431, 43mtbird 328 . . . . 5 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → ¬ (𝑔𝑧) ∈ ran 𝐹)
4529, 44eldifd 3864 . . . 4 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → (𝑔𝑧) ∈ (𝐵 ∖ ran 𝐹))
4645ex 416 . . 3 (𝜑 → (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) → (𝑔𝑧) ∈ (𝐵 ∖ ran 𝐹)))
4721, 46syl5bi 245 . 2 (𝜑 → (𝑔 ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} → (𝑔𝑧) ∈ (𝐵 ∖ ran 𝐹)))
48 hashf1lem1.5 . . . . . . 7 (𝜑𝐹:𝐴1-1𝐵)
49 f1f 6585 . . . . . . 7 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
5048, 49syl 17 . . . . . 6 (𝜑𝐹:𝐴𝐵)
5150adantr 484 . . . . 5 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → 𝐹:𝐴𝐵)
52 vex 3404 . . . . . . . 8 𝑥 ∈ V
5325, 52f1osn 6670 . . . . . . 7 {⟨𝑧, 𝑥⟩}:{𝑧}–1-1-onto→{𝑥}
54 f1of 6631 . . . . . . 7 ({⟨𝑧, 𝑥⟩}:{𝑧}–1-1-onto→{𝑥} → {⟨𝑧, 𝑥⟩}:{𝑧}⟶{𝑥})
5553, 54ax-mp 5 . . . . . 6 {⟨𝑧, 𝑥⟩}:{𝑧}⟶{𝑥}
56 eldifi 4027 . . . . . . . 8 (𝑥 ∈ (𝐵 ∖ ran 𝐹) → 𝑥𝐵)
5756adantl 485 . . . . . . 7 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → 𝑥𝐵)
5857snssd 4707 . . . . . 6 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → {𝑥} ⊆ 𝐵)
59 fss 6532 . . . . . 6 (({⟨𝑧, 𝑥⟩}:{𝑧}⟶{𝑥} ∧ {𝑥} ⊆ 𝐵) → {⟨𝑧, 𝑥⟩}:{𝑧}⟶𝐵)
6055, 58, 59sylancr 590 . . . . 5 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → {⟨𝑧, 𝑥⟩}:{𝑧}⟶𝐵)
61 res0 5839 . . . . . . 7 (𝐹 ↾ ∅) = ∅
62 res0 5839 . . . . . . 7 ({⟨𝑧, 𝑥⟩} ↾ ∅) = ∅
6361, 62eqtr4i 2765 . . . . . 6 (𝐹 ↾ ∅) = ({⟨𝑧, 𝑥⟩} ↾ ∅)
64 disjsn 4612 . . . . . . . . 9 ((𝐴 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝐴)
6530, 64sylibr 237 . . . . . . . 8 (𝜑 → (𝐴 ∩ {𝑧}) = ∅)
6665adantr 484 . . . . . . 7 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐴 ∩ {𝑧}) = ∅)
6766reseq2d 5835 . . . . . 6 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ↾ (𝐴 ∩ {𝑧})) = (𝐹 ↾ ∅))
6866reseq2d 5835 . . . . . 6 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ({⟨𝑧, 𝑥⟩} ↾ (𝐴 ∩ {𝑧})) = ({⟨𝑧, 𝑥⟩} ↾ ∅))
6963, 67, 683eqtr4a 2800 . . . . 5 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ↾ (𝐴 ∩ {𝑧})) = ({⟨𝑧, 𝑥⟩} ↾ (𝐴 ∩ {𝑧})))
70 fresaunres1 6562 . . . . 5 ((𝐹:𝐴𝐵 ∧ {⟨𝑧, 𝑥⟩}:{𝑧}⟶𝐵 ∧ (𝐹 ↾ (𝐴 ∩ {𝑧})) = ({⟨𝑧, 𝑥⟩} ↾ (𝐴 ∩ {𝑧}))) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹)
7151, 60, 69, 70syl3anc 1372 . . . 4 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹)
72 f1f1orn 6642 . . . . . . . . 9 (𝐹:𝐴1-1𝐵𝐹:𝐴1-1-onto→ran 𝐹)
7348, 72syl 17 . . . . . . . 8 (𝜑𝐹:𝐴1-1-onto→ran 𝐹)
7473adantr 484 . . . . . . 7 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → 𝐹:𝐴1-1-onto→ran 𝐹)
7553a1i 11 . . . . . . 7 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → {⟨𝑧, 𝑥⟩}:{𝑧}–1-1-onto→{𝑥})
76 eldifn 4028 . . . . . . . . 9 (𝑥 ∈ (𝐵 ∖ ran 𝐹) → ¬ 𝑥 ∈ ran 𝐹)
7776adantl 485 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ¬ 𝑥 ∈ ran 𝐹)
78 disjsn 4612 . . . . . . . 8 ((ran 𝐹 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ ran 𝐹)
7977, 78sylibr 237 . . . . . . 7 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (ran 𝐹 ∩ {𝑥}) = ∅)
80 f1oun 6650 . . . . . . 7 (((𝐹:𝐴1-1-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}))
8174, 75, 66, 79, 80syl22anc 838 . . . . . 6 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}))
82 f1of1 6630 . . . . . 6 ((𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→(ran 𝐹 ∪ {𝑥}))
8381, 82syl 17 . . . . 5 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→(ran 𝐹 ∪ {𝑥}))
8451frnd 6523 . . . . . 6 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ran 𝐹𝐵)
8584, 58unssd 4086 . . . . 5 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (ran 𝐹 ∪ {𝑥}) ⊆ 𝐵)
86 f1ss 6591 . . . . 5 (((𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→(ran 𝐹 ∪ {𝑥}) ∧ (ran 𝐹 ∪ {𝑥}) ⊆ 𝐵) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1𝐵)
8783, 85, 86syl2anc 587 . . . 4 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1𝐵)
8850, 4fexd 7013 . . . . . . 7 (𝜑𝐹 ∈ V)
8988adantr 484 . . . . . 6 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → 𝐹 ∈ V)
90 snex 5308 . . . . . 6 {⟨𝑧, 𝑥⟩} ∈ V
91 unexg 7503 . . . . . 6 ((𝐹 ∈ V ∧ {⟨𝑧, 𝑥⟩} ∈ V) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ V)
9289, 90, 91sylancl 589 . . . . 5 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ V)
93 reseq1 5829 . . . . . . . 8 (𝑓 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) → (𝑓𝐴) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴))
9493eqeq1d 2741 . . . . . . 7 (𝑓 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) → ((𝑓𝐴) = 𝐹 ↔ ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹))
95 f1eq1 6580 . . . . . . 7 (𝑓 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) → (𝑓:(𝐴 ∪ {𝑧})–1-1𝐵 ↔ (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1𝐵))
9694, 95anbi12d 634 . . . . . 6 (𝑓 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) → (((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵) ↔ (((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹 ∧ (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1𝐵)))
9796elabg 3576 . . . . 5 ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ V → ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ↔ (((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹 ∧ (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1𝐵)))
9892, 97syl 17 . . . 4 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ↔ (((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹 ∧ (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1𝐵)))
9971, 87, 98mpbir2and 713 . . 3 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)})
10099ex 416 . 2 (𝜑 → (𝑥 ∈ (𝐵 ∖ ran 𝐹) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)}))
10121anbi1i 627 . . 3 ((𝑔 ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) ↔ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)))
102 simprlr 780 . . . . . . 7 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → 𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)
103 f1fn 6586 . . . . . . 7 (𝑔:(𝐴 ∪ {𝑧})–1-1𝐵𝑔 Fn (𝐴 ∪ {𝑧}))
104102, 103syl 17 . . . . . 6 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → 𝑔 Fn (𝐴 ∪ {𝑧}))
10581adantrl 716 . . . . . . 7 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}))
106 f1ofn 6632 . . . . . . 7 ((𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) Fn (𝐴 ∪ {𝑧}))
107105, 106syl 17 . . . . . 6 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) Fn (𝐴 ∪ {𝑧}))
108 eqfnfv 6822 . . . . . 6 ((𝑔 Fn (𝐴 ∪ {𝑧}) ∧ (𝐹 ∪ {⟨𝑧, 𝑥⟩}) Fn (𝐴 ∪ {𝑧})) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ ∀𝑦 ∈ (𝐴 ∪ {𝑧})(𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦)))
109104, 107, 108syl2anc 587 . . . . 5 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ ∀𝑦 ∈ (𝐴 ∪ {𝑧})(𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦)))
110 fvres 6706 . . . . . . . . . . 11 (𝑦𝐴 → ((𝑔𝐴)‘𝑦) = (𝑔𝑦))
111110eqcomd 2745 . . . . . . . . . 10 (𝑦𝐴 → (𝑔𝑦) = ((𝑔𝐴)‘𝑦))
112 simprll 779 . . . . . . . . . . 11 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝑔𝐴) = 𝐹)
113112fveq1d 6689 . . . . . . . . . 10 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ((𝑔𝐴)‘𝑦) = (𝐹𝑦))
114111, 113sylan9eqr 2796 . . . . . . . . 9 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → (𝑔𝑦) = (𝐹𝑦))
11548ad2antrr 726 . . . . . . . . . . 11 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → 𝐹:𝐴1-1𝐵)
116 f1fn 6586 . . . . . . . . . . 11 (𝐹:𝐴1-1𝐵𝐹 Fn 𝐴)
117115, 116syl 17 . . . . . . . . . 10 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → 𝐹 Fn 𝐴)
11825, 52fnsn 6408 . . . . . . . . . . 11 {⟨𝑧, 𝑥⟩} Fn {𝑧}
119118a1i 11 . . . . . . . . . 10 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → {⟨𝑧, 𝑥⟩} Fn {𝑧})
12065ad2antrr 726 . . . . . . . . . 10 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → (𝐴 ∩ {𝑧}) = ∅)
121 simpr 488 . . . . . . . . . 10 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → 𝑦𝐴)
122117, 119, 120, 121fvun1d 6774 . . . . . . . . 9 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) = (𝐹𝑦))
123114, 122eqtr4d 2777 . . . . . . . 8 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦))
124123ralrimiva 3097 . . . . . . 7 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ∀𝑦𝐴 (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦))
125124biantrurd 536 . . . . . 6 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (∀𝑦 ∈ {𝑧} (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ (∀𝑦𝐴 (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ∧ ∀𝑦 ∈ {𝑧} (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦))))
126 ralunb 4091 . . . . . 6 (∀𝑦 ∈ (𝐴 ∪ {𝑧})(𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ (∀𝑦𝐴 (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ∧ ∀𝑦 ∈ {𝑧} (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦)))
127125, 126bitr4di 292 . . . . 5 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (∀𝑦 ∈ {𝑧} (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ ∀𝑦 ∈ (𝐴 ∪ {𝑧})(𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦)))
12850fdmd 6526 . . . . . . . . . . 11 (𝜑 → dom 𝐹 = 𝐴)
129128eleq2d 2819 . . . . . . . . . 10 (𝜑 → (𝑧 ∈ dom 𝐹𝑧𝐴))
13030, 129mtbird 328 . . . . . . . . 9 (𝜑 → ¬ 𝑧 ∈ dom 𝐹)
131130adantr 484 . . . . . . . 8 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ¬ 𝑧 ∈ dom 𝐹)
132 fsnunfv 6972 . . . . . . . 8 ((𝑧 ∈ V ∧ 𝑥 ∈ V ∧ ¬ 𝑧 ∈ dom 𝐹) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧) = 𝑥)
13325, 52, 131, 132mp3an12i 1466 . . . . . . 7 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧) = 𝑥)
134133eqeq2d 2750 . . . . . 6 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ((𝑔𝑧) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧) ↔ (𝑔𝑧) = 𝑥))
135 fveq2 6687 . . . . . . . 8 (𝑦 = 𝑧 → (𝑔𝑦) = (𝑔𝑧))
136 fveq2 6687 . . . . . . . 8 (𝑦 = 𝑧 → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧))
137135, 136eqeq12d 2755 . . . . . . 7 (𝑦 = 𝑧 → ((𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ (𝑔𝑧) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧)))
13825, 137ralsn 4582 . . . . . 6 (∀𝑦 ∈ {𝑧} (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ (𝑔𝑧) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧))
139 eqcom 2746 . . . . . 6 (𝑥 = (𝑔𝑧) ↔ (𝑔𝑧) = 𝑥)
140134, 138, 1393bitr4g 317 . . . . 5 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (∀𝑦 ∈ {𝑧} (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ 𝑥 = (𝑔𝑧)))
141109, 127, 1403bitr2d 310 . . . 4 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ 𝑥 = (𝑔𝑧)))
142141ex 416 . . 3 (𝜑 → ((((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ 𝑥 = (𝑔𝑧))))
143101, 142syl5bi 245 . 2 (𝜑 → ((𝑔 ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ 𝑥 = (𝑔𝑧))))
14413, 15, 47, 100, 143en3d 8605 1 (𝜑 → {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ≈ (𝐵 ∖ ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1542  wcel 2114  {cab 2717  wral 3054  Vcvv 3400  cdif 3850  cun 3851  cin 3852  wss 3853  c0 4221  {csn 4526  cop 4532   class class class wbr 5040  dom cdm 5535  ran crn 5536  cres 5537  cima 5538   Fn wfn 6345  wf 6346  1-1wf1 6347  1-1-ontowf1o 6349  cfv 6350  (class class class)co 7183  m cmap 8450  cen 8565  Fincfn 8568  1c1 10629   + caddc 10631  cle 10767  chash 13795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7492
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4807  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-tr 5147  df-id 5439  df-eprel 5444  df-po 5452  df-so 5453  df-fr 5493  df-we 5495  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-ord 6186  df-on 6187  df-lim 6188  df-suc 6189  df-iota 6308  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-ov 7186  df-oprab 7187  df-mpo 7188  df-om 7613  df-1o 8144  df-map 8452  df-en 8569  df-fin 8572
This theorem is referenced by: (None)
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