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Theorem f1ocpbllem 17579

Description: Lemma for f1ocpbl 17580. (Contributed by Mario Carneiro, 24-Feb-2015.)

**Hypothesis**
Ref	Expression
f1ocpbl.f	⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝑋)

**Assertion**
Ref	Expression
f1ocpbllem	⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (((𝐹‘𝐴) = (𝐹‘𝐶) ∧ (𝐹‘𝐵) = (𝐹‘𝐷)) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))

**Proof of Theorem f1ocpbllem**
Step	Hyp	Ref	Expression
1		f1ocpbl.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝑋)
2		f1of1 6821	. . . . 5 ⊢ (𝐹:𝑉–1-1-onto→𝑋 → 𝐹:𝑉–1-1→𝑋)
3	1, 2	syl 18	. . . 4 ⊢ (𝜑 → 𝐹:𝑉–1-1→𝑋)
4	3	3ad2ant1 1151	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐹:𝑉–1-1→𝑋)
5		simp2l 1218	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐴 ∈ 𝑉)
6		simp3l 1220	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐶 ∈ 𝑉)
7		f1fveq 7262	. . 3 ⊢ ((𝐹:𝑉–1-1→𝑋 ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐹‘𝐴) = (𝐹‘𝐶) ↔ 𝐴 = 𝐶))
8	4, 5, 6, 7	syl12anc 849	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((𝐹‘𝐴) = (𝐹‘𝐶) ↔ 𝐴 = 𝐶))
9		simp2r 1219	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐵 ∈ 𝑉)
10		simp3r 1221	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐷 ∈ 𝑉)
11		f1fveq 7262	. . 3 ⊢ ((𝐹:𝑉–1-1→𝑋 ∧ (𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((𝐹‘𝐵) = (𝐹‘𝐷) ↔ 𝐵 = 𝐷))
12	4, 9, 10, 11	syl12anc 849	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((𝐹‘𝐵) = (𝐹‘𝐷) ↔ 𝐵 = 𝐷))
13	8, 12	anbi12d 643	1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (((𝐹‘𝐴) = (𝐹‘𝐶) ∧ (𝐹‘𝐵) = (𝐹‘𝐷)) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 –1-1→wf1 6535 –1-1-onto→wf1o 6537 ‘cfv 6538

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5258 ax-nul 5270 ax-pr 5406

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-ne 2959 df-ral 3080 df-rex 3090 df-rab 3417 df-v 3457 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-if 4489 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-br 5111 df-opab 5175 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-f1o 6545 df-fv 6546

This theorem is referenced by: f1ocpbl 17580 f1olecpbl 17582

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