MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hashf1lem1 Structured version   Visualization version   GIF version

Theorem hashf1lem1 13440
Description: Lemma for hashf1 13442. (Contributed by Mario Carneiro, 17-Apr-2015.)
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
hashf1lem1 (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 hashf1lem1
Dummy variables 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1f 6283 . . . . . 6 (𝑓:(𝐴 ∪ {𝑧})–1-1𝐵𝑓:(𝐴 ∪ {𝑧})⟶𝐵)
21adantl 473 . . . . 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 8245 . . . . . . 7 {𝑧} ∈ Fin
6 unfi 8434 . . . . . . 7 ((𝐴 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝐴 ∪ {𝑧}) ∈ Fin)
74, 5, 6sylancl 580 . . . . . 6 (𝜑 → (𝐴 ∪ {𝑧}) ∈ Fin)
83, 7elmapd 8074 . . . . 5 (𝜑 → (𝑓 ∈ (𝐵𝑚 (𝐴 ∪ {𝑧})) ↔ 𝑓:(𝐴 ∪ {𝑧})⟶𝐵))
92, 8syl5ibr 237 . . . 4 (𝜑 → (((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵) → 𝑓 ∈ (𝐵𝑚 (𝐴 ∪ {𝑧}))))
109abssdv 3836 . . 3 (𝜑 → {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ⊆ (𝐵𝑚 (𝐴 ∪ {𝑧})))
11 ovex 6874 . . 3 (𝐵𝑚 (𝐴 ∪ {𝑧})) ∈ V
12 ssexg 4965 . . 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 580 . 2 (𝜑 → {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ∈ V)
14 difexg 4969 . . 3 (𝐵 ∈ Fin → (𝐵 ∖ ran 𝐹) ∈ V)
153, 14syl 17 . 2 (𝜑 → (𝐵 ∖ ran 𝐹) ∈ V)
16 vex 3353 . . . 4 𝑔 ∈ V
17 reseq1 5559 . . . . . 6 (𝑓 = 𝑔 → (𝑓𝐴) = (𝑔𝐴))
1817eqeq1d 2767 . . . . 5 (𝑓 = 𝑔 → ((𝑓𝐴) = 𝐹 ↔ (𝑔𝐴) = 𝐹))
19 f1eq1 6278 . . . . 5 (𝑓 = 𝑔 → (𝑓:(𝐴 ∪ {𝑧})–1-1𝐵𝑔:(𝐴 ∪ {𝑧})–1-1𝐵))
2018, 19anbi12d 624 . . . 4 (𝑓 = 𝑔 → (((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵) ↔ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)))
2116, 20elab 3504 . . 3 (𝑔 ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ↔ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵))
22 f1f 6283 . . . . . . 7 (𝑔:(𝐴 ∪ {𝑧})–1-1𝐵𝑔:(𝐴 ∪ {𝑧})⟶𝐵)
2322ad2antll 720 . . . . . 6 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → 𝑔:(𝐴 ∪ {𝑧})⟶𝐵)
24 ssun2 3939 . . . . . . 7 {𝑧} ⊆ (𝐴 ∪ {𝑧})
25 vex 3353 . . . . . . . 8 𝑧 ∈ V
2625snss 4470 . . . . . . 7 (𝑧 ∈ (𝐴 ∪ {𝑧}) ↔ {𝑧} ⊆ (𝐴 ∪ {𝑧}))
2724, 26mpbir 222 . . . . . 6 𝑧 ∈ (𝐴 ∪ {𝑧})
28 ffvelrn 6547 . . . . . 6 ((𝑔:(𝐴 ∪ {𝑧})⟶𝐵𝑧 ∈ (𝐴 ∪ {𝑧})) → (𝑔𝑧) ∈ 𝐵)
2923, 27, 28sylancl 580 . . . . 5 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → (𝑔𝑧) ∈ 𝐵)
30 hashf1lem2.3 . . . . . . 7 (𝜑 → ¬ 𝑧𝐴)
3130adantr 472 . . . . . 6 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → ¬ 𝑧𝐴)
32 df-ima 5290 . . . . . . . . 9 (𝑔𝐴) = ran (𝑔𝐴)
33 simprl 787 . . . . . . . . . 10 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → (𝑔𝐴) = 𝐹)
3433rneqd 5521 . . . . . . . . 9 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → ran (𝑔𝐴) = ran 𝐹)
3532, 34syl5eq 2811 . . . . . . . 8 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → (𝑔𝐴) = ran 𝐹)
3635eleq2d 2830 . . . . . . 7 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → ((𝑔𝑧) ∈ (𝑔𝐴) ↔ (𝑔𝑧) ∈ ran 𝐹))
37 simprr 789 . . . . . . . 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 3938 . . . . . . . . 9 𝐴 ⊆ (𝐴 ∪ {𝑧})
4039a1i 11 . . . . . . . 8 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → 𝐴 ⊆ (𝐴 ∪ {𝑧}))
41 f1elima 6712 . . . . . . . 8 ((𝑔:(𝐴 ∪ {𝑧})–1-1𝐵𝑧 ∈ (𝐴 ∪ {𝑧}) ∧ 𝐴 ⊆ (𝐴 ∪ {𝑧})) → ((𝑔𝑧) ∈ (𝑔𝐴) ↔ 𝑧𝐴))
4237, 38, 40, 41syl3anc 1490 . . . . . . 7 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → ((𝑔𝑧) ∈ (𝑔𝐴) ↔ 𝑧𝐴))
4336, 42bitr3d 272 . . . . . 6 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → ((𝑔𝑧) ∈ ran 𝐹𝑧𝐴))
4431, 43mtbird 316 . . . . 5 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → ¬ (𝑔𝑧) ∈ ran 𝐹)
4529, 44eldifd 3743 . . . 4 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → (𝑔𝑧) ∈ (𝐵 ∖ ran 𝐹))
4645ex 401 . . 3 (𝜑 → (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) → (𝑔𝑧) ∈ (𝐵 ∖ ran 𝐹)))
4721, 46syl5bi 233 . 2 (𝜑 → (𝑔 ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} → (𝑔𝑧) ∈ (𝐵 ∖ ran 𝐹)))
48 hashf1lem1.5 . . . . . . 7 (𝜑𝐹:𝐴1-1𝐵)
49 f1f 6283 . . . . . . 7 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
5048, 49syl 17 . . . . . 6 (𝜑𝐹:𝐴𝐵)
5150adantr 472 . . . . 5 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → 𝐹:𝐴𝐵)
52 vex 3353 . . . . . . . 8 𝑥 ∈ V
5325, 52f1osn 6359 . . . . . . 7 {⟨𝑧, 𝑥⟩}:{𝑧}–1-1-onto→{𝑥}
54 f1of 6320 . . . . . . 7 ({⟨𝑧, 𝑥⟩}:{𝑧}–1-1-onto→{𝑥} → {⟨𝑧, 𝑥⟩}:{𝑧}⟶{𝑥})
5553, 54ax-mp 5 . . . . . 6 {⟨𝑧, 𝑥⟩}:{𝑧}⟶{𝑥}
56 eldifi 3894 . . . . . . . 8 (𝑥 ∈ (𝐵 ∖ ran 𝐹) → 𝑥𝐵)
5756adantl 473 . . . . . . 7 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → 𝑥𝐵)
5857snssd 4494 . . . . . 6 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → {𝑥} ⊆ 𝐵)
59 fss 6236 . . . . . 6 (({⟨𝑧, 𝑥⟩}:{𝑧}⟶{𝑥} ∧ {𝑥} ⊆ 𝐵) → {⟨𝑧, 𝑥⟩}:{𝑧}⟶𝐵)
6055, 58, 59sylancr 581 . . . . 5 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → {⟨𝑧, 𝑥⟩}:{𝑧}⟶𝐵)
61 res0 5569 . . . . . . 7 (𝐹 ↾ ∅) = ∅
62 res0 5569 . . . . . . 7 ({⟨𝑧, 𝑥⟩} ↾ ∅) = ∅
6361, 62eqtr4i 2790 . . . . . 6 (𝐹 ↾ ∅) = ({⟨𝑧, 𝑥⟩} ↾ ∅)
64 disjsn 4402 . . . . . . . . 9 ((𝐴 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝐴)
6530, 64sylibr 225 . . . . . . . 8 (𝜑 → (𝐴 ∩ {𝑧}) = ∅)
6665adantr 472 . . . . . . 7 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐴 ∩ {𝑧}) = ∅)
6766reseq2d 5565 . . . . . 6 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ↾ (𝐴 ∩ {𝑧})) = (𝐹 ↾ ∅))
6866reseq2d 5565 . . . . . 6 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ({⟨𝑧, 𝑥⟩} ↾ (𝐴 ∩ {𝑧})) = ({⟨𝑧, 𝑥⟩} ↾ ∅))
6963, 67, 683eqtr4a 2825 . . . . 5 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ↾ (𝐴 ∩ {𝑧})) = ({⟨𝑧, 𝑥⟩} ↾ (𝐴 ∩ {𝑧})))
70 fresaunres1 6259 . . . . 5 ((𝐹:𝐴𝐵 ∧ {⟨𝑧, 𝑥⟩}:{𝑧}⟶𝐵 ∧ (𝐹 ↾ (𝐴 ∩ {𝑧})) = ({⟨𝑧, 𝑥⟩} ↾ (𝐴 ∩ {𝑧}))) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹)
7151, 60, 69, 70syl3anc 1490 . . . 4 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹)
72 f1f1orn 6331 . . . . . . . . 9 (𝐹:𝐴1-1𝐵𝐹:𝐴1-1-onto→ran 𝐹)
7348, 72syl 17 . . . . . . . 8 (𝜑𝐹:𝐴1-1-onto→ran 𝐹)
7473adantr 472 . . . . . . 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 3895 . . . . . . . . 9 (𝑥 ∈ (𝐵 ∖ ran 𝐹) → ¬ 𝑥 ∈ ran 𝐹)
7776adantl 473 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ¬ 𝑥 ∈ ran 𝐹)
78 disjsn 4402 . . . . . . . 8 ((ran 𝐹 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ ran 𝐹)
7977, 78sylibr 225 . . . . . . 7 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (ran 𝐹 ∩ {𝑥}) = ∅)
80 f1oun 6339 . . . . . . 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 867 . . . . . 6 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}))
82 f1of1 6319 . . . . . 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 6230 . . . . . 6 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ran 𝐹𝐵)
8584, 58unssd 3951 . . . . 5 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (ran 𝐹 ∪ {𝑥}) ⊆ 𝐵)
86 f1ss 6288 . . . . 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 579 . . . 4 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1𝐵)
88 fex 6682 . . . . . . . 8 ((𝐹:𝐴𝐵𝐴 ∈ Fin) → 𝐹 ∈ V)
8950, 4, 88syl2anc 579 . . . . . . 7 (𝜑𝐹 ∈ V)
9089adantr 472 . . . . . 6 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → 𝐹 ∈ V)
91 snex 5064 . . . . . 6 {⟨𝑧, 𝑥⟩} ∈ V
92 unexg 7157 . . . . . 6 ((𝐹 ∈ V ∧ {⟨𝑧, 𝑥⟩} ∈ V) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ V)
9390, 91, 92sylancl 580 . . . . 5 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ V)
94 reseq1 5559 . . . . . . . 8 (𝑓 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) → (𝑓𝐴) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴))
9594eqeq1d 2767 . . . . . . 7 (𝑓 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) → ((𝑓𝐴) = 𝐹 ↔ ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹))
96 f1eq1 6278 . . . . . . 7 (𝑓 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) → (𝑓:(𝐴 ∪ {𝑧})–1-1𝐵 ↔ (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1𝐵))
9795, 96anbi12d 624 . . . . . 6 (𝑓 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) → (((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵) ↔ (((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹 ∧ (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1𝐵)))
9897elabg 3505 . . . . 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)))
9993, 98syl 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)))
10071, 87, 99mpbir2and 704 . . 3 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)})
101100ex 401 . 2 (𝜑 → (𝑥 ∈ (𝐵 ∖ ran 𝐹) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)}))
10221anbi1i 617 . . 3 ((𝑔 ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) ↔ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)))
103 simprlr 798 . . . . . . 7 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → 𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)
104 f1fn 6284 . . . . . . 7 (𝑔:(𝐴 ∪ {𝑧})–1-1𝐵𝑔 Fn (𝐴 ∪ {𝑧}))
105103, 104syl 17 . . . . . 6 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → 𝑔 Fn (𝐴 ∪ {𝑧}))
10681adantrl 707 . . . . . . 7 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}))
107 f1ofn 6321 . . . . . . 7 ((𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) Fn (𝐴 ∪ {𝑧}))
108106, 107syl 17 . . . . . 6 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) Fn (𝐴 ∪ {𝑧}))
109 eqfnfv 6501 . . . . . 6 ((𝑔 Fn (𝐴 ∪ {𝑧}) ∧ (𝐹 ∪ {⟨𝑧, 𝑥⟩}) Fn (𝐴 ∪ {𝑧})) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ ∀𝑦 ∈ (𝐴 ∪ {𝑧})(𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦)))
110105, 108, 109syl2anc 579 . . . . 5 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ ∀𝑦 ∈ (𝐴 ∪ {𝑧})(𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦)))
111 fvres 6394 . . . . . . . . . . 11 (𝑦𝐴 → ((𝑔𝐴)‘𝑦) = (𝑔𝑦))
112111eqcomd 2771 . . . . . . . . . 10 (𝑦𝐴 → (𝑔𝑦) = ((𝑔𝐴)‘𝑦))
113 simprll 797 . . . . . . . . . . 11 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝑔𝐴) = 𝐹)
114113fveq1d 6377 . . . . . . . . . 10 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ((𝑔𝐴)‘𝑦) = (𝐹𝑦))
115112, 114sylan9eqr 2821 . . . . . . . . 9 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → (𝑔𝑦) = (𝐹𝑦))
11648ad2antrr 717 . . . . . . . . . . 11 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → 𝐹:𝐴1-1𝐵)
117 f1fn 6284 . . . . . . . . . . 11 (𝐹:𝐴1-1𝐵𝐹 Fn 𝐴)
118116, 117syl 17 . . . . . . . . . 10 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → 𝐹 Fn 𝐴)
11925, 52fnsn 6125 . . . . . . . . . . 11 {⟨𝑧, 𝑥⟩} Fn {𝑧}
120119a1i 11 . . . . . . . . . 10 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → {⟨𝑧, 𝑥⟩} Fn {𝑧})
12165ad2antrr 717 . . . . . . . . . 10 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → (𝐴 ∩ {𝑧}) = ∅)
122 simpr 477 . . . . . . . . . 10 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → 𝑦𝐴)
123 fvun1 6458 . . . . . . . . . 10 ((𝐹 Fn 𝐴 ∧ {⟨𝑧, 𝑥⟩} Fn {𝑧} ∧ ((𝐴 ∩ {𝑧}) = ∅ ∧ 𝑦𝐴)) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) = (𝐹𝑦))
124118, 120, 121, 122, 123syl112anc 1493 . . . . . . . . 9 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) = (𝐹𝑦))
125115, 124eqtr4d 2802 . . . . . . . 8 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦))
126125ralrimiva 3113 . . . . . . 7 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ∀𝑦𝐴 (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦))
127126biantrurd 528 . . . . . 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))))
128 ralunb 3956 . . . . . 6 (∀𝑦 ∈ (𝐴 ∪ {𝑧})(𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ (∀𝑦𝐴 (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ∧ ∀𝑦 ∈ {𝑧} (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦)))
129127, 128syl6bbr 280 . . . . 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)))
13025a1i 11 . . . . . . . 8 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → 𝑧 ∈ V)
13152a1i 11 . . . . . . . 8 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → 𝑥 ∈ V)
13250fdmd 6232 . . . . . . . . . . 11 (𝜑 → dom 𝐹 = 𝐴)
133132eleq2d 2830 . . . . . . . . . 10 (𝜑 → (𝑧 ∈ dom 𝐹𝑧𝐴))
13430, 133mtbird 316 . . . . . . . . 9 (𝜑 → ¬ 𝑧 ∈ dom 𝐹)
135134adantr 472 . . . . . . . 8 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ¬ 𝑧 ∈ dom 𝐹)
136 fsnunfv 6646 . . . . . . . 8 ((𝑧 ∈ V ∧ 𝑥 ∈ V ∧ ¬ 𝑧 ∈ dom 𝐹) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧) = 𝑥)
137130, 131, 135, 136syl3anc 1490 . . . . . . 7 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧) = 𝑥)
138137eqeq2d 2775 . . . . . 6 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ((𝑔𝑧) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧) ↔ (𝑔𝑧) = 𝑥))
139 fveq2 6375 . . . . . . . 8 (𝑦 = 𝑧 → (𝑔𝑦) = (𝑔𝑧))
140 fveq2 6375 . . . . . . . 8 (𝑦 = 𝑧 → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧))
141139, 140eqeq12d 2780 . . . . . . 7 (𝑦 = 𝑧 → ((𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ (𝑔𝑧) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧)))
14225, 141ralsn 4379 . . . . . 6 (∀𝑦 ∈ {𝑧} (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ (𝑔𝑧) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧))
143 eqcom 2772 . . . . . 6 (𝑥 = (𝑔𝑧) ↔ (𝑔𝑧) = 𝑥)
144138, 142, 1433bitr4g 305 . . . . 5 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (∀𝑦 ∈ {𝑧} (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ 𝑥 = (𝑔𝑧)))
145110, 129, 1443bitr2d 298 . . . 4 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ 𝑥 = (𝑔𝑧)))
146145ex 401 . . 3 (𝜑 → ((((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ 𝑥 = (𝑔𝑧))))
147102, 146syl5bi 233 . 2 (𝜑 → ((𝑔 ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ 𝑥 = (𝑔𝑧))))
14813, 15, 47, 101, 147en3d 8197 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 197  wa 384   = wceq 1652  wcel 2155  {cab 2751  wral 3055  Vcvv 3350  cdif 3729  cun 3730  cin 3731  wss 3732  c0 4079  {csn 4334  cop 4340   class class class wbr 4809  dom cdm 5277  ran crn 5278  cres 5279  cima 5280   Fn wfn 6063  wf 6064  1-1wf1 6065  1-1-ontowf1o 6067  cfv 6068  (class class class)co 6842  𝑚 cmap 8060  cen 8157  Fincfn 8160  1c1 10190   + caddc 10192  cle 10329  chash 13321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-map 8062  df-en 8161  df-fin 8164
This theorem is referenced by:  hashf1lem2  13441
  Copyright terms: Public domain W3C validator