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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 32176. (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 2742 | . 2 ⊢ ∅ = (𝐸‘∅) |
3 | fvprc 6838 | . 2 ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑋) = ∅) | |
4 | oveqprc.z | . . . 4 ⊢ 𝑍 = (𝑋𝑂𝑌) | |
5 | oveqprc.r | . . . . 5 ⊢ Rel dom 𝑂 | |
6 | 5 | ovprc1 7400 | . . . 4 ⊢ (¬ 𝑋 ∈ V → (𝑋𝑂𝑌) = ∅) |
7 | 4, 6 | eqtrid 2785 | . . 3 ⊢ (¬ 𝑋 ∈ V → 𝑍 = ∅) |
8 | 7 | fveq2d 6850 | . 2 ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑍) = (𝐸‘∅)) |
9 | 2, 3, 8 | 3eqtr4a 2799 | 1 ⊢ (¬ 𝑋 ∈ V → (𝐸‘𝑋) = (𝐸‘𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3447 ∅c0 4286 dom cdm 5637 Rel wrel 5642 ‘cfv 6500 (class class class)co 7361 |
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 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-xp 5643 df-rel 5644 df-dm 5647 df-iota 6452 df-fv 6508 df-ov 7364 |
This theorem is referenced by: setsnid 17089 ressbas 17126 resseqnbas 17130 tnglem 24019 resvlem 32176 |
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