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Mirrors > Home > MPE Home > Th. List > ercpbllem | Structured version Visualization version GIF version |
Description: Lemma for ercpbl 16209. (Contributed by Mario Carneiro, 24-Feb-2015.) |
Ref | Expression |
---|---|
ercpbl.r | ⊢ (𝜑 → ∼ Er 𝑉) |
ercpbl.v | ⊢ (𝜑 → 𝑉 ∈ V) |
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 ∈ V) | |
3 | ercpbl.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) | |
4 | 1, 2, 3 | divsfval 16207 | . . 3 ⊢ (𝜑 → (𝐹‘𝐴) = [𝐴] ∼ ) |
5 | 1, 2, 3 | divsfval 16207 | . . 3 ⊢ (𝜑 → (𝐹‘𝐵) = [𝐵] ∼ ) |
6 | 4, 5 | eqeq12d 2637 | . 2 ⊢ (𝜑 → ((𝐹‘𝐴) = (𝐹‘𝐵) ↔ [𝐴] ∼ = [𝐵] ∼ )) |
7 | ercpbllem.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
8 | 1, 7 | erth 7791 | . 2 ⊢ (𝜑 → (𝐴 ∼ 𝐵 ↔ [𝐴] ∼ = [𝐵] ∼ )) |
9 | 6, 8 | bitr4d 271 | 1 ⊢ (𝜑 → ((𝐹‘𝐴) = (𝐹‘𝐵) ↔ 𝐴 ∼ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 ∈ wcel 1990 Vcvv 3200 class class class wbr 4653 ↦ cmpt 4729 ‘cfv 5888 Er wer 7739 [cec 7740 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fv 5896 df-er 7742 df-ec 7744 |
This theorem is referenced by: ercpbl 16209 erlecpbl 16210 |
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