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Mirrors > Home > MPE Home > Th. List > ercpbllem | Structured version Visualization version GIF version |
Description: Lemma for ercpbl 17504. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by AV, 12-Jul-2024.) |
Ref | Expression |
---|---|
ercpbl.r | ⊢ (𝜑 → ∼ Er 𝑉) |
ercpbl.v | ⊢ (𝜑 → 𝑉 ∈ 𝑊) |
ercpbl.f | ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) |
ercpbllem.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
Ref | Expression |
---|---|
ercpbllem | ⊢ (𝜑 → ((𝐹‘𝐴) = (𝐹‘𝐵) ↔ 𝐴 ∼ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ercpbl.r | . . . 4 ⊢ (𝜑 → ∼ Er 𝑉) | |
2 | ercpbl.v | . . . 4 ⊢ (𝜑 → 𝑉 ∈ 𝑊) | |
3 | ercpbl.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) | |
4 | 1, 2, 3 | divsfval 17502 | . . 3 ⊢ (𝜑 → (𝐹‘𝐴) = [𝐴] ∼ ) |
5 | 1, 2, 3 | divsfval 17502 | . . 3 ⊢ (𝜑 → (𝐹‘𝐵) = [𝐵] ∼ ) |
6 | 4, 5 | eqeq12d 2742 | . 2 ⊢ (𝜑 → ((𝐹‘𝐴) = (𝐹‘𝐵) ↔ [𝐴] ∼ = [𝐵] ∼ )) |
7 | ercpbllem.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
8 | 1, 7 | erth 8754 | . 2 ⊢ (𝜑 → (𝐴 ∼ 𝐵 ↔ [𝐴] ∼ = [𝐵] ∼ )) |
9 | 6, 8 | bitr4d 282 | 1 ⊢ (𝜑 → ((𝐹‘𝐴) = (𝐹‘𝐵) ↔ 𝐴 ∼ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 class class class wbr 5141 ↦ cmpt 5224 ‘cfv 6537 Er wer 8702 [cec 8703 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fv 6545 df-er 8705 df-ec 8707 |
This theorem is referenced by: ercpbl 17504 erlecpbl 17505 |
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