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Mirrors > Home > MPE Home > Th. List > oveqprc | Structured version Visualization version GIF version |
Description: Lemma for showing the equality of values for functions like slot extractors 𝐸 at a proper class. Extracted from several former proofs of lemmas like resvlem 33337. (Contributed by AV, 31-Oct-2024.) |
Ref | Expression |
---|---|
oveqprc.e | ⊢ (𝐸‘∅) = ∅ |
oveqprc.z | ⊢ 𝑍 = (𝑋𝑂𝑌) |
oveqprc.r | ⊢ Rel dom 𝑂 |
Ref | Expression |
---|---|
oveqprc | ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑋) = (𝐸‘𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveqprc.e | . . 3 ⊢ (𝐸‘∅) = ∅ | |
2 | 1 | eqcomi 2744 | . 2 ⊢ ∅ = (𝐸‘∅) |
3 | fvprc 6899 | . 2 ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑋) = ∅) | |
4 | oveqprc.z | . . . 4 ⊢ 𝑍 = (𝑋𝑂𝑌) | |
5 | oveqprc.r | . . . . 5 ⊢ Rel dom 𝑂 | |
6 | 5 | ovprc1 7470 | . . . 4 ⊢ (¬ 𝑋 ∈ V → (𝑋𝑂𝑌) = ∅) |
7 | 4, 6 | eqtrid 2787 | . . 3 ⊢ (¬ 𝑋 ∈ V → 𝑍 = ∅) |
8 | 7 | fveq2d 6911 | . 2 ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑍) = (𝐸‘∅)) |
9 | 2, 3, 8 | 3eqtr4a 2801 | 1 ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑋) = (𝐸‘𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 dom cdm 5689 Rel wrel 5694 ‘cfv 6563 (class class class)co 7431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-dm 5699 df-iota 6516 df-fv 6571 df-ov 7434 |
This theorem is referenced by: setsnid 17243 ressbas 17280 resseqnbas 17287 tnglem 24669 resvlem 33337 |
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