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| Description: Lemma for ercpbl 17595. (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 17593 | . . 3 ⊢ (𝜑 → (𝐹‘𝐴) = [𝐴] ∼ ) | 
| 5 | 1, 2, 3 | divsfval 17593 | . . 3 ⊢ (𝜑 → (𝐹‘𝐵) = [𝐵] ∼ ) | 
| 6 | 4, 5 | eqeq12d 2752 | . 2 ⊢ (𝜑 → ((𝐹‘𝐴) = (𝐹‘𝐵) ↔ [𝐴] ∼ = [𝐵] ∼ )) | 
| 7 | ercpbllem.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 8 | 1, 7 | erth 8797 | . 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 206 = wceq 1539 ∈ wcel 2107 class class class wbr 5142 ↦ cmpt 5224 ‘cfv 6560 Er wer 8743 [cec 8744 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fv 6568 df-er 8746 df-ec 8748 | 
| This theorem is referenced by: ercpbl 17595 erlecpbl 17596 | 
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