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Mirrors > Home > MPE Home > Th. List > mapsnf1o3 | Structured version Visualization version GIF version |
Description: Explicit bijection in the reverse of mapsnf1o2 8890. (Contributed by Stefan O'Rear, 24-Mar-2015.) |
Ref | Expression |
---|---|
mapsncnv.s | ⊢ 𝑆 = {𝑋} |
mapsncnv.b | ⊢ 𝐵 ∈ V |
mapsncnv.x | ⊢ 𝑋 ∈ V |
mapsnf1o3.f | ⊢ 𝐹 = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦})) |
Ref | Expression |
---|---|
mapsnf1o3 | ⊢ 𝐹:𝐵–1-1-onto→(𝐵 ↑m 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapsncnv.s | . . . 4 ⊢ 𝑆 = {𝑋} | |
2 | mapsncnv.b | . . . 4 ⊢ 𝐵 ∈ V | |
3 | mapsncnv.x | . . . 4 ⊢ 𝑋 ∈ V | |
4 | eqid 2726 | . . . 4 ⊢ (𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)) = (𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)) | |
5 | 1, 2, 3, 4 | mapsnf1o2 8890 | . . 3 ⊢ (𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)):(𝐵 ↑m 𝑆)–1-1-onto→𝐵 |
6 | f1ocnv 6839 | . . 3 ⊢ ((𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)):(𝐵 ↑m 𝑆)–1-1-onto→𝐵 → ◡(𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)):𝐵–1-1-onto→(𝐵 ↑m 𝑆)) | |
7 | 5, 6 | ax-mp 5 | . 2 ⊢ ◡(𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)):𝐵–1-1-onto→(𝐵 ↑m 𝑆) |
8 | mapsnf1o3.f | . . . 4 ⊢ 𝐹 = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦})) | |
9 | 1, 2, 3, 4 | mapsncnv 8889 | . . . 4 ⊢ ◡(𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)) = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦})) |
10 | 8, 9 | eqtr4i 2757 | . . 3 ⊢ 𝐹 = ◡(𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)) |
11 | f1oeq1 6815 | . . 3 ⊢ (𝐹 = ◡(𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)) → (𝐹:𝐵–1-1-onto→(𝐵 ↑m 𝑆) ↔ ◡(𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)):𝐵–1-1-onto→(𝐵 ↑m 𝑆))) | |
12 | 10, 11 | ax-mp 5 | . 2 ⊢ (𝐹:𝐵–1-1-onto→(𝐵 ↑m 𝑆) ↔ ◡(𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)):𝐵–1-1-onto→(𝐵 ↑m 𝑆)) |
13 | 7, 12 | mpbir 230 | 1 ⊢ 𝐹:𝐵–1-1-onto→(𝐵 ↑m 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1533 ∈ wcel 2098 Vcvv 3468 {csn 4623 ↦ cmpt 5224 × cxp 5667 ◡ccnv 5668 –1-1-onto→wf1o 6536 ‘cfv 6537 (class class class)co 7405 ↑m cmap 8822 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-map 8824 |
This theorem is referenced by: coe1f2 22083 coe1add 22138 evls1rhmlem 22195 evl1sca 22208 pf1ind 22229 ismrer1 37219 |
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