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Mirrors > Home > MPE Home > Th. List > f1ocpbllem | Structured version Visualization version GIF version |
Description: Lemma for f1ocpbl 17506. (Contributed by Mario Carneiro, 24-Feb-2015.) |
Ref | Expression |
---|---|
f1ocpbl.f | ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝑋) |
Ref | Expression |
---|---|
f1ocpbllem | ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (((𝐹‘𝐴) = (𝐹‘𝐶) ∧ (𝐹‘𝐵) = (𝐹‘𝐷)) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ocpbl.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝑋) | |
2 | f1of1 6833 | . . . . 5 ⊢ (𝐹:𝑉–1-1-onto→𝑋 → 𝐹:𝑉–1-1→𝑋) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹:𝑉–1-1→𝑋) |
4 | 3 | 3ad2ant1 1130 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐹:𝑉–1-1→𝑋) |
5 | simp2l 1196 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐴 ∈ 𝑉) | |
6 | simp3l 1198 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐶 ∈ 𝑉) | |
7 | f1fveq 7268 | . . 3 ⊢ ((𝐹:𝑉–1-1→𝑋 ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐹‘𝐴) = (𝐹‘𝐶) ↔ 𝐴 = 𝐶)) | |
8 | 4, 5, 6, 7 | syl12anc 835 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((𝐹‘𝐴) = (𝐹‘𝐶) ↔ 𝐴 = 𝐶)) |
9 | simp2r 1197 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐵 ∈ 𝑉) | |
10 | simp3r 1199 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐷 ∈ 𝑉) | |
11 | f1fveq 7268 | . . 3 ⊢ ((𝐹:𝑉–1-1→𝑋 ∧ (𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((𝐹‘𝐵) = (𝐹‘𝐷) ↔ 𝐵 = 𝐷)) | |
12 | 4, 9, 10, 11 | syl12anc 835 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((𝐹‘𝐵) = (𝐹‘𝐷) ↔ 𝐵 = 𝐷)) |
13 | 8, 12 | anbi12d 630 | 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 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 –1-1→wf1 6540 –1-1-onto→wf1o 6542 ‘cfv 6543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-f1o 6550 df-fv 6551 |
This theorem is referenced by: f1ocpbl 17506 f1olecpbl 17508 |
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