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Mirrors > Home > MPE Home > Th. List > Mathboxes > fnwe2lem1 | Structured version Visualization version GIF version |
Description: Lemma for fnwe2 39660. Substitution in well-ordering hypothesis. (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))} |
fnwe2.s | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)}) |
Ref | Expression |
---|---|
fnwe2lem1 | ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnwe2.s | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)}) | |
2 | 1 | ralrimiva 3184 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)}) |
3 | fveq2 6672 | . . . . . . 7 ⊢ (𝑎 = 𝑥 → (𝐹‘𝑎) = (𝐹‘𝑥)) | |
4 | 3 | csbeq1d 3889 | . . . . . 6 ⊢ (𝑎 = 𝑥 → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 = ⦋(𝐹‘𝑥) / 𝑧⦌𝑆) |
5 | fvex 6685 | . . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V | |
6 | fnwe2.su | . . . . . . 7 ⊢ (𝑧 = (𝐹‘𝑥) → 𝑆 = 𝑈) | |
7 | 5, 6 | csbie 3920 | . . . . . 6 ⊢ ⦋(𝐹‘𝑥) / 𝑧⦌𝑆 = 𝑈 |
8 | 4, 7 | syl6eq 2874 | . . . . 5 ⊢ (𝑎 = 𝑥 → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 = 𝑈) |
9 | 3 | eqeq2d 2834 | . . . . . 6 ⊢ (𝑎 = 𝑥 → ((𝐹‘𝑦) = (𝐹‘𝑎) ↔ (𝐹‘𝑦) = (𝐹‘𝑥))) |
10 | 9 | rabbidv 3482 | . . . . 5 ⊢ (𝑎 = 𝑥 → {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)} = {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)}) |
11 | 8, 10 | weeq12d 39647 | . . . 4 ⊢ (𝑎 = 𝑥 → (⦋(𝐹‘𝑎) / 𝑧⦌𝑆 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)} ↔ 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)})) |
12 | 11 | cbvralvw 3451 | . . 3 ⊢ (∀𝑎 ∈ 𝐴 ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)} ↔ ∀𝑥 ∈ 𝐴 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)}) |
13 | 2, 12 | sylibr 236 | . 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)}) |
14 | 13 | r19.21bi 3210 | 1 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ∀wral 3140 {crab 3144 ⦋csb 3885 class class class wbr 5068 {copab 5130 We wwe 5515 ‘cfv 6357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-nul 5212 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-iota 6316 df-fv 6365 |
This theorem is referenced by: fnwe2lem2 39658 fnwe2lem3 39659 |
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