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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 8460. (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 2823 | . . . 4 ⊢ (𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)) = (𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)) | |
5 | 1, 2, 3, 4 | mapsnf1o2 8460 | . . 3 ⊢ (𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)):(𝐵 ↑m 𝑆)–1-1-onto→𝐵 |
6 | f1ocnv 6629 | . . 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 8459 | . . . 4 ⊢ ◡(𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)) = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦})) |
10 | 8, 9 | eqtr4i 2849 | . . 3 ⊢ 𝐹 = ◡(𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)) |
11 | f1oeq1 6606 | . . 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 233 | 1 ⊢ 𝐹:𝐵–1-1-onto→(𝐵 ↑m 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∈ wcel 2114 Vcvv 3496 {csn 4569 ↦ cmpt 5148 × cxp 5555 ◡ccnv 5556 –1-1-onto→wf1o 6356 ‘cfv 6357 (class class class)co 7158 ↑m cmap 8408 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-map 8410 |
This theorem is referenced by: coe1f2 20379 coe1add 20434 evls1rhmlem 20486 evl1sca 20499 pf1ind 20520 ismrer1 35118 |
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