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Mirrors > Home > MPE Home > Th. List > ercpbllem | Structured version Visualization version GIF version |
Description: Lemma for ercpbl 17432. (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 17430 | . . 3 ⊢ (𝜑 → (𝐹‘𝐴) = [𝐴] ∼ ) |
5 | 1, 2, 3 | divsfval 17430 | . . 3 ⊢ (𝜑 → (𝐹‘𝐵) = [𝐵] ∼ ) |
6 | 4, 5 | eqeq12d 2753 | . 2 ⊢ (𝜑 → ((𝐹‘𝐴) = (𝐹‘𝐵) ↔ [𝐴] ∼ = [𝐵] ∼ )) |
7 | ercpbllem.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
8 | 1, 7 | erth 8698 | . 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 1542 ∈ wcel 2107 class class class wbr 5106 ↦ cmpt 5189 ‘cfv 6497 Er wer 8646 [cec 8647 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fv 6505 df-er 8649 df-ec 8651 |
This theorem is referenced by: ercpbl 17432 erlecpbl 17433 |
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