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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 8835. (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 2733 | . . . 4 ⊢ (𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)) = (𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)) | |
5 | 1, 2, 3, 4 | mapsnf1o2 8835 | . . 3 ⊢ (𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)):(𝐵 ↑m 𝑆)–1-1-onto→𝐵 |
6 | f1ocnv 6797 | . . 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 8834 | . . . 4 ⊢ ◡(𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)) = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦})) |
10 | 8, 9 | eqtr4i 2764 | . . 3 ⊢ 𝐹 = ◡(𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)) |
11 | f1oeq1 6773 | . . 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 1542 ∈ wcel 2107 Vcvv 3444 {csn 4587 ↦ cmpt 5189 × cxp 5632 ◡ccnv 5633 –1-1-onto→wf1o 6496 ‘cfv 6497 (class class class)co 7358 ↑m cmap 8768 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-map 8770 |
This theorem is referenced by: coe1f2 21596 coe1add 21651 evls1rhmlem 21703 evl1sca 21716 pf1ind 21737 ismrer1 36343 |
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