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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 8575. (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 2737 | . . . 4 ⊢ (𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)) = (𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)) | |
5 | 1, 2, 3, 4 | mapsnf1o2 8575 | . . 3 ⊢ (𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)):(𝐵 ↑m 𝑆)–1-1-onto→𝐵 |
6 | f1ocnv 6673 | . . 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 8574 | . . . 4 ⊢ ◡(𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)) = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦})) |
10 | 8, 9 | eqtr4i 2768 | . . 3 ⊢ 𝐹 = ◡(𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)) |
11 | f1oeq1 6649 | . . 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 234 | 1 ⊢ 𝐹:𝐵–1-1-onto→(𝐵 ↑m 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1543 ∈ wcel 2110 Vcvv 3408 {csn 4541 ↦ cmpt 5135 × cxp 5549 ◡ccnv 5550 –1-1-onto→wf1o 6379 ‘cfv 6380 (class class class)co 7213 ↑m cmap 8508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-1st 7761 df-2nd 7762 df-map 8510 |
This theorem is referenced by: coe1f2 21130 coe1add 21185 evls1rhmlem 21237 evl1sca 21250 pf1ind 21271 ismrer1 35733 |
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