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Mirrors > Home > MPE Home > Th. List > Mathboxes > fnwe2val | Structured version Visualization version GIF version |
Description: Lemma for fnwe2 42370. Substitute variables. (Contributed by Stefan O'Rear, 19-Jan-2015.) |
Ref | Expression |
---|---|
fnwe2.su | ⊢ (𝑧 = (𝐹‘𝑥) → 𝑆 = 𝑈) |
fnwe2.t | ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑈𝑦))} |
Ref | Expression |
---|---|
fnwe2val | ⊢ (𝑎𝑇𝑏 ↔ ((𝐹‘𝑎)𝑅(𝐹‘𝑏) ∨ ((𝐹‘𝑎) = (𝐹‘𝑏) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3472 | . 2 ⊢ 𝑎 ∈ V | |
2 | vex 3472 | . 2 ⊢ 𝑏 ∈ V | |
3 | fveq2 6885 | . . . 4 ⊢ (𝑥 = 𝑎 → (𝐹‘𝑥) = (𝐹‘𝑎)) | |
4 | fveq2 6885 | . . . 4 ⊢ (𝑦 = 𝑏 → (𝐹‘𝑦) = (𝐹‘𝑏)) | |
5 | 3, 4 | breqan12d 5157 | . . 3 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ↔ (𝐹‘𝑎)𝑅(𝐹‘𝑏))) |
6 | 3, 4 | eqeqan12d 2740 | . . . 4 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐹‘𝑎) = (𝐹‘𝑏))) |
7 | simpl 482 | . . . . 5 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → 𝑥 = 𝑎) | |
8 | fvex 6898 | . . . . . . . 8 ⊢ (𝐹‘𝑥) ∈ V | |
9 | fnwe2.su | . . . . . . . 8 ⊢ (𝑧 = (𝐹‘𝑥) → 𝑆 = 𝑈) | |
10 | 8, 9 | csbie 3924 | . . . . . . 7 ⊢ ⦋(𝐹‘𝑥) / 𝑧⦌𝑆 = 𝑈 |
11 | 3 | csbeq1d 3892 | . . . . . . 7 ⊢ (𝑥 = 𝑎 → ⦋(𝐹‘𝑥) / 𝑧⦌𝑆 = ⦋(𝐹‘𝑎) / 𝑧⦌𝑆) |
12 | 10, 11 | eqtr3id 2780 | . . . . . 6 ⊢ (𝑥 = 𝑎 → 𝑈 = ⦋(𝐹‘𝑎) / 𝑧⦌𝑆) |
13 | 12 | adantr 480 | . . . . 5 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → 𝑈 = ⦋(𝐹‘𝑎) / 𝑧⦌𝑆) |
14 | simpr 484 | . . . . 5 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → 𝑦 = 𝑏) | |
15 | 7, 13, 14 | breq123d 5155 | . . . 4 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝑥𝑈𝑦 ↔ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏)) |
16 | 6, 15 | anbi12d 630 | . . 3 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑈𝑦) ↔ ((𝐹‘𝑎) = (𝐹‘𝑏) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏))) |
17 | 5, 16 | orbi12d 915 | . 2 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑈𝑦)) ↔ ((𝐹‘𝑎)𝑅(𝐹‘𝑏) ∨ ((𝐹‘𝑎) = (𝐹‘𝑏) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏)))) |
18 | fnwe2.t | . 2 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑈𝑦))} | |
19 | 1, 2, 17, 18 | braba 5530 | 1 ⊢ (𝑎𝑇𝑏 ↔ ((𝐹‘𝑎)𝑅(𝐹‘𝑏) ∨ ((𝐹‘𝑎) = (𝐹‘𝑏) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 844 = wceq 1533 ⦋csb 3888 class class class wbr 5141 {copab 5203 ‘cfv 6537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-iota 6489 df-fv 6545 |
This theorem is referenced by: fnwe2lem2 42368 fnwe2lem3 42369 |
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