|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > fparlem1 | Structured version Visualization version GIF version | ||
| Description: Lemma for fpar 8141. (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 6925 | . . . . . 6 ⊢ (𝑦 ∈ (V × V) → ((1st ↾ (V × V))‘𝑦) = (1st ‘𝑦)) | |
| 2 | 1 | eqeq1d 2739 | . . . . 5 ⊢ (𝑦 ∈ (V × V) → (((1st ↾ (V × V))‘𝑦) = 𝑥 ↔ (1st ‘𝑦) = 𝑥)) | 
| 3 | vex 3484 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
| 4 | 3 | elsn2 4665 | . . . . . 6 ⊢ ((1st ‘𝑦) ∈ {𝑥} ↔ (1st ‘𝑦) = 𝑥) | 
| 5 | fvex 6919 | . . . . . . 7 ⊢ (2nd ‘𝑦) ∈ V | |
| 6 | 5 | biantru 529 | . . . . . 6 ⊢ ((1st ‘𝑦) ∈ {𝑥} ↔ ((1st ‘𝑦) ∈ {𝑥} ∧ (2nd ‘𝑦) ∈ V)) | 
| 7 | 4, 6 | bitr3i 277 | . . . . 5 ⊢ ((1st ‘𝑦) = 𝑥 ↔ ((1st ‘𝑦) ∈ {𝑥} ∧ (2nd ‘𝑦) ∈ V)) | 
| 8 | 2, 7 | bitrdi 287 | . . . 4 ⊢ (𝑦 ∈ (V × V) → (((1st ↾ (V × V))‘𝑦) = 𝑥 ↔ ((1st ‘𝑦) ∈ {𝑥} ∧ (2nd ‘𝑦) ∈ V))) | 
| 9 | 8 | pm5.32i 574 | . . 3 ⊢ ((𝑦 ∈ (V × V) ∧ ((1st ↾ (V × V))‘𝑦) = 𝑥) ↔ (𝑦 ∈ (V × V) ∧ ((1st ‘𝑦) ∈ {𝑥} ∧ (2nd ‘𝑦) ∈ V))) | 
| 10 | f1stres 8038 | . . . 4 ⊢ (1st ↾ (V × V)):(V × V)⟶V | |
| 11 | ffn 6736 | . . . 4 ⊢ ((1st ↾ (V × V)):(V × V)⟶V → (1st ↾ (V × V)) Fn (V × V)) | |
| 12 | fniniseg 7080 | . . . 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 8049 | . . 3 ⊢ (𝑦 ∈ ({𝑥} × V) ↔ (𝑦 ∈ (V × V) ∧ ((1st ‘𝑦) ∈ {𝑥} ∧ (2nd ‘𝑦) ∈ V))) | |
| 15 | 9, 13, 14 | 3bitr4i 303 | . 2 ⊢ (𝑦 ∈ (◡(1st ↾ (V × V)) “ {𝑥}) ↔ 𝑦 ∈ ({𝑥} × V)) | 
| 16 | 15 | eqriv 2734 | 1 ⊢ (◡(1st ↾ (V × V)) “ {𝑥}) = ({𝑥} × V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 {csn 4626 × cxp 5683 ◡ccnv 5684 ↾ cres 5687 “ cima 5688 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 1st c1st 8012 2nd c2nd 8013 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-1st 8014 df-2nd 8015 | 
| This theorem is referenced by: fparlem3 8139 | 
| Copyright terms: Public domain | W3C validator |