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Mirrors > Home > MPE Home > Th. List > Mathboxes > fnwe2lem1 | Structured version Visualization version GIF version |
Description: Lemma for fnwe2 36641. 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 2949 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)}) |
3 | fveq2 6103 | . . . . . . 7 ⊢ (𝑎 = 𝑥 → (𝐹‘𝑎) = (𝐹‘𝑥)) | |
4 | 3 | csbeq1d 3506 | . . . . . 6 ⊢ (𝑎 = 𝑥 → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 = ⦋(𝐹‘𝑥) / 𝑧⦌𝑆) |
5 | fvex 6113 | . . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V | |
6 | fnwe2.su | . . . . . . 7 ⊢ (𝑧 = (𝐹‘𝑥) → 𝑆 = 𝑈) | |
7 | 5, 6 | csbie 3525 | . . . . . 6 ⊢ ⦋(𝐹‘𝑥) / 𝑧⦌𝑆 = 𝑈 |
8 | 4, 7 | syl6eq 2660 | . . . . 5 ⊢ (𝑎 = 𝑥 → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 = 𝑈) |
9 | 3 | eqeq2d 2620 | . . . . . 6 ⊢ (𝑎 = 𝑥 → ((𝐹‘𝑦) = (𝐹‘𝑎) ↔ (𝐹‘𝑦) = (𝐹‘𝑥))) |
10 | 9 | rabbidv 3164 | . . . . 5 ⊢ (𝑎 = 𝑥 → {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)} = {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)}) |
11 | 8, 10 | weeq12d 36628 | . . . 4 ⊢ (𝑎 = 𝑥 → (⦋(𝐹‘𝑎) / 𝑧⦌𝑆 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)} ↔ 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)})) |
12 | 11 | cbvralv 3147 | . . 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 223 | . 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)}) |
14 | 13 | r19.21bi 2916 | 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 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 ⦋csb 3499 class class class wbr 4583 {copab 4642 We wwe 4996 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-iota 5768 df-fv 5812 |
This theorem is referenced by: fnwe2lem2 36639 fnwe2lem3 36640 |
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