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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 8933. (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 2735 | . . . 4 ⊢ (𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)) = (𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)) | |
5 | 1, 2, 3, 4 | mapsnf1o2 8933 | . . 3 ⊢ (𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)):(𝐵 ↑m 𝑆)–1-1-onto→𝐵 |
6 | f1ocnv 6861 | . . 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 8932 | . . . 4 ⊢ ◡(𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)) = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦})) |
10 | 8, 9 | eqtr4i 2766 | . . 3 ⊢ 𝐹 = ◡(𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)) |
11 | f1oeq1 6837 | . . 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 231 | 1 ⊢ 𝐹:𝐵–1-1-onto→(𝐵 ↑m 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 ∈ wcel 2106 Vcvv 3478 {csn 4631 ↦ cmpt 5231 × cxp 5687 ◡ccnv 5688 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-map 8867 |
This theorem is referenced by: coe1f2 22227 coe1add 22283 evls1rhmlem 22341 evl1sca 22354 pf1ind 22375 ismrer1 37825 |
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