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Mirrors > Home > MPE Home > Th. List > fparlem1 | Structured version Visualization version GIF version |
Description: Lemma for fpar 8127. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
fparlem1 | ⊢ (◡(1st ↾ (V × V)) “ {𝑥}) = ({𝑥} × V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvres 6921 | . . . . . 6 ⊢ (𝑦 ∈ (V × V) → ((1st ↾ (V × V))‘𝑦) = (1st ‘𝑦)) | |
2 | 1 | eqeq1d 2730 | . . . . 5 ⊢ (𝑦 ∈ (V × V) → (((1st ↾ (V × V))‘𝑦) = 𝑥 ↔ (1st ‘𝑦) = 𝑥)) |
3 | vex 3477 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
4 | 3 | elsn2 4672 | . . . . . 6 ⊢ ((1st ‘𝑦) ∈ {𝑥} ↔ (1st ‘𝑦) = 𝑥) |
5 | fvex 6915 | . . . . . . 7 ⊢ (2nd ‘𝑦) ∈ V | |
6 | 5 | biantru 528 | . . . . . 6 ⊢ ((1st ‘𝑦) ∈ {𝑥} ↔ ((1st ‘𝑦) ∈ {𝑥} ∧ (2nd ‘𝑦) ∈ V)) |
7 | 4, 6 | bitr3i 276 | . . . . 5 ⊢ ((1st ‘𝑦) = 𝑥 ↔ ((1st ‘𝑦) ∈ {𝑥} ∧ (2nd ‘𝑦) ∈ V)) |
8 | 2, 7 | bitrdi 286 | . . . 4 ⊢ (𝑦 ∈ (V × V) → (((1st ↾ (V × V))‘𝑦) = 𝑥 ↔ ((1st ‘𝑦) ∈ {𝑥} ∧ (2nd ‘𝑦) ∈ V))) |
9 | 8 | pm5.32i 573 | . . 3 ⊢ ((𝑦 ∈ (V × V) ∧ ((1st ↾ (V × V))‘𝑦) = 𝑥) ↔ (𝑦 ∈ (V × V) ∧ ((1st ‘𝑦) ∈ {𝑥} ∧ (2nd ‘𝑦) ∈ V))) |
10 | f1stres 8023 | . . . 4 ⊢ (1st ↾ (V × V)):(V × V)⟶V | |
11 | ffn 6727 | . . . 4 ⊢ ((1st ↾ (V × V)):(V × V)⟶V → (1st ↾ (V × V)) Fn (V × V)) | |
12 | fniniseg 7074 | . . . 4 ⊢ ((1st ↾ (V × V)) Fn (V × V) → (𝑦 ∈ (◡(1st ↾ (V × V)) “ {𝑥}) ↔ (𝑦 ∈ (V × V) ∧ ((1st ↾ (V × V))‘𝑦) = 𝑥))) | |
13 | 10, 11, 12 | mp2b 10 | . . 3 ⊢ (𝑦 ∈ (◡(1st ↾ (V × V)) “ {𝑥}) ↔ (𝑦 ∈ (V × V) ∧ ((1st ↾ (V × V))‘𝑦) = 𝑥)) |
14 | elxp7 8034 | . . 3 ⊢ (𝑦 ∈ ({𝑥} × V) ↔ (𝑦 ∈ (V × V) ∧ ((1st ‘𝑦) ∈ {𝑥} ∧ (2nd ‘𝑦) ∈ V))) | |
15 | 9, 13, 14 | 3bitr4i 302 | . 2 ⊢ (𝑦 ∈ (◡(1st ↾ (V × V)) “ {𝑥}) ↔ 𝑦 ∈ ({𝑥} × V)) |
16 | 15 | eqriv 2725 | 1 ⊢ (◡(1st ↾ (V × V)) “ {𝑥}) = ({𝑥} × V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3473 {csn 4632 × cxp 5680 ◡ccnv 5681 ↾ cres 5684 “ cima 5685 Fn wfn 6548 ⟶wf 6549 ‘cfv 6553 1st c1st 7997 2nd c2nd 7998 |
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-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-1st 7999 df-2nd 8000 |
This theorem is referenced by: fparlem3 8125 |
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