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Mirrors > Home > MPE Home > Th. List > fvpr1OLD | Structured version Visualization version GIF version |
Description: Obsolete version of fvpr1 7193 as of 26-Sep-2024. (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
fvpr1.1 | ⊢ 𝐴 ∈ V |
fvpr1.2 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
fvpr1OLD | ⊢ (𝐴 ≠ 𝐵 → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 4631 | . . . 4 ⊢ {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩}) | |
2 | 1 | fveq1i 6892 | . . 3 ⊢ ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = (({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})‘𝐴) |
3 | necom 2994 | . . . 4 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴) | |
4 | fvunsn 7179 | . . . 4 ⊢ (𝐵 ≠ 𝐴 → (({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})‘𝐴) = ({⟨𝐴, 𝐶⟩}‘𝐴)) | |
5 | 3, 4 | sylbi 216 | . . 3 ⊢ (𝐴 ≠ 𝐵 → (({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})‘𝐴) = ({⟨𝐴, 𝐶⟩}‘𝐴)) |
6 | 2, 5 | eqtrid 2784 | . 2 ⊢ (𝐴 ≠ 𝐵 → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = ({⟨𝐴, 𝐶⟩}‘𝐴)) |
7 | fvpr1.1 | . . 3 ⊢ 𝐴 ∈ V | |
8 | fvpr1.2 | . . 3 ⊢ 𝐶 ∈ V | |
9 | 7, 8 | fvsn 7181 | . 2 ⊢ ({⟨𝐴, 𝐶⟩}‘𝐴) = 𝐶 |
10 | 6, 9 | eqtrdi 2788 | 1 ⊢ (𝐴 ≠ 𝐵 → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 Vcvv 3474 ∪ cun 3946 {csn 4628 {cpr 4630 ⟨cop 4634 ‘cfv 6543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-res 5688 df-iota 6495 df-fun 6545 df-fv 6551 |
This theorem is referenced by: (None) |
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