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Mirrors > Home > MPE Home > Th. List > Mathboxes > fnwe2val | Structured version Visualization version GIF version |
Description: Lemma for fnwe2 43042. 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 3482 | . 2 ⊢ 𝑎 ∈ V | |
2 | vex 3482 | . 2 ⊢ 𝑏 ∈ V | |
3 | fveq2 6907 | . . . 4 ⊢ (𝑥 = 𝑎 → (𝐹‘𝑥) = (𝐹‘𝑎)) | |
4 | fveq2 6907 | . . . 4 ⊢ (𝑦 = 𝑏 → (𝐹‘𝑦) = (𝐹‘𝑏)) | |
5 | 3, 4 | breqan12d 5164 | . . 3 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ↔ (𝐹‘𝑎)𝑅(𝐹‘𝑏))) |
6 | 3, 4 | eqeqan12d 2749 | . . . 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 6920 | . . . . . . . 8 ⊢ (𝐹‘𝑥) ∈ V | |
9 | fnwe2.su | . . . . . . . 8 ⊢ (𝑧 = (𝐹‘𝑥) → 𝑆 = 𝑈) | |
10 | 8, 9 | csbie 3944 | . . . . . . 7 ⊢ ⦋(𝐹‘𝑥) / 𝑧⦌𝑆 = 𝑈 |
11 | 3 | csbeq1d 3912 | . . . . . . 7 ⊢ (𝑥 = 𝑎 → ⦋(𝐹‘𝑥) / 𝑧⦌𝑆 = ⦋(𝐹‘𝑎) / 𝑧⦌𝑆) |
12 | 10, 11 | eqtr3id 2789 | . . . . . 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 5162 | . . . 4 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝑥𝑈𝑦 ↔ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏)) |
16 | 6, 15 | anbi12d 632 | . . 3 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑈𝑦) ↔ ((𝐹‘𝑎) = (𝐹‘𝑏) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏))) |
17 | 5, 16 | orbi12d 918 | . 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 5547 | 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 206 ∧ wa 395 ∨ wo 847 = wceq 1537 ⦋csb 3908 class class class wbr 5148 {copab 5210 ‘cfv 6563 |
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-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-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 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-iota 6516 df-fv 6571 |
This theorem is referenced by: fnwe2lem2 43040 fnwe2lem3 43041 |
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