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Mirrors > Home > MPE Home > Th. List > ercpbllem | Structured version Visualization version GIF version |
Description: Lemma for ercpbl 16822. (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 16820 | . . 3 ⊢ (𝜑 → (𝐹‘𝐴) = [𝐴] ∼ ) |
5 | 1, 2, 3 | divsfval 16820 | . . 3 ⊢ (𝜑 → (𝐹‘𝐵) = [𝐵] ∼ ) |
6 | 4, 5 | eqeq12d 2837 | . 2 ⊢ (𝜑 → ((𝐹‘𝐴) = (𝐹‘𝐵) ↔ [𝐴] ∼ = [𝐵] ∼ )) |
7 | ercpbllem.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
8 | 1, 7 | erth 8338 | . 2 ⊢ (𝜑 → (𝐴 ∼ 𝐵 ↔ [𝐴] ∼ = [𝐵] ∼ )) |
9 | 6, 8 | bitr4d 284 | 1 ⊢ (𝜑 → ((𝐹‘𝐴) = (𝐹‘𝐵) ↔ 𝐴 ∼ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 Vcvv 3494 class class class wbr 5066 ↦ cmpt 5146 ‘cfv 6355 Er wer 8286 [cec 8287 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fv 6363 df-er 8289 df-ec 8291 |
This theorem is referenced by: ercpbl 16822 erlecpbl 16823 |
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