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Mirrors > Home > MPE Home > Th. List > fparlem2 | Structured version Visualization version GIF version |
Description: Lemma for fpar 7168. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
fparlem2 | ⊢ (◡(2nd ↾ (V × V)) “ {𝑦}) = (V × {𝑦}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvres 6117 | . . . . . 6 ⊢ (𝑥 ∈ (V × V) → ((2nd ↾ (V × V))‘𝑥) = (2nd ‘𝑥)) | |
2 | 1 | eqeq1d 2612 | . . . . 5 ⊢ (𝑥 ∈ (V × V) → (((2nd ↾ (V × V))‘𝑥) = 𝑦 ↔ (2nd ‘𝑥) = 𝑦)) |
3 | vex 3176 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
4 | 3 | elsn2 4158 | . . . . . 6 ⊢ ((2nd ‘𝑥) ∈ {𝑦} ↔ (2nd ‘𝑥) = 𝑦) |
5 | fvex 6113 | . . . . . . 7 ⊢ (1st ‘𝑥) ∈ V | |
6 | 5 | biantrur 526 | . . . . . 6 ⊢ ((2nd ‘𝑥) ∈ {𝑦} ↔ ((1st ‘𝑥) ∈ V ∧ (2nd ‘𝑥) ∈ {𝑦})) |
7 | 4, 6 | bitr3i 265 | . . . . 5 ⊢ ((2nd ‘𝑥) = 𝑦 ↔ ((1st ‘𝑥) ∈ V ∧ (2nd ‘𝑥) ∈ {𝑦})) |
8 | 2, 7 | syl6bb 275 | . . . 4 ⊢ (𝑥 ∈ (V × V) → (((2nd ↾ (V × V))‘𝑥) = 𝑦 ↔ ((1st ‘𝑥) ∈ V ∧ (2nd ‘𝑥) ∈ {𝑦}))) |
9 | 8 | pm5.32i 667 | . . 3 ⊢ ((𝑥 ∈ (V × V) ∧ ((2nd ↾ (V × V))‘𝑥) = 𝑦) ↔ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ V ∧ (2nd ‘𝑥) ∈ {𝑦}))) |
10 | f2ndres 7082 | . . . 4 ⊢ (2nd ↾ (V × V)):(V × V)⟶V | |
11 | ffn 5958 | . . . 4 ⊢ ((2nd ↾ (V × V)):(V × V)⟶V → (2nd ↾ (V × V)) Fn (V × V)) | |
12 | fniniseg 6246 | . . . 4 ⊢ ((2nd ↾ (V × V)) Fn (V × V) → (𝑥 ∈ (◡(2nd ↾ (V × V)) “ {𝑦}) ↔ (𝑥 ∈ (V × V) ∧ ((2nd ↾ (V × V))‘𝑥) = 𝑦))) | |
13 | 10, 11, 12 | mp2b 10 | . . 3 ⊢ (𝑥 ∈ (◡(2nd ↾ (V × V)) “ {𝑦}) ↔ (𝑥 ∈ (V × V) ∧ ((2nd ↾ (V × V))‘𝑥) = 𝑦)) |
14 | elxp7 7092 | . . 3 ⊢ (𝑥 ∈ (V × {𝑦}) ↔ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ V ∧ (2nd ‘𝑥) ∈ {𝑦}))) | |
15 | 9, 13, 14 | 3bitr4i 291 | . 2 ⊢ (𝑥 ∈ (◡(2nd ↾ (V × V)) “ {𝑦}) ↔ 𝑥 ∈ (V × {𝑦})) |
16 | 15 | eqriv 2607 | 1 ⊢ (◡(2nd ↾ (V × V)) “ {𝑦}) = (V × {𝑦}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 {csn 4125 × cxp 5036 ◡ccnv 5037 ↾ cres 5040 “ cima 5041 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 1st c1st 7057 2nd c2nd 7058 |
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-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 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-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-1st 7059 df-2nd 7060 |
This theorem is referenced by: fparlem4 7167 |
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