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Theorem mapsnf1o3 8822

Description: Explicit bijection in the reverse of mapsnf1o2 8821. (Contributed by Stefan O'Rear, 24-Mar-2015.)

**Hypotheses**
Ref	Expression
mapsncnv.s	⊢ 𝑆 = {𝑋}
mapsncnv.b	⊢ 𝐵 ∈ V
mapsncnv.x	⊢ 𝑋 ∈ V
mapsnf1o3.f	⊢ 𝐹 = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦}))

**Assertion**
Ref	Expression
mapsnf1o3	⊢ 𝐹:𝐵–1-1-onto→(𝐵 ↑_m 𝑆)

Distinct variable groups: 𝑦,𝐵 𝑦,𝑆 𝑦,𝑋 Allowed substitution hint: 𝐹(𝑦)

Proof of Theorem mapsnf1o3 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mapsncnv.s	. . . 4 ⊢ 𝑆 = {𝑋}
2		mapsncnv.b	. . . 4 ⊢ 𝐵 ∈ V
3		mapsncnv.x	. . . 4 ⊢ 𝑋 ∈ V
4		eqid 2729	. . . 4 ⊢ (𝑥 ∈ (𝐵 ↑_m 𝑆) ↦ (𝑥‘𝑋)) = (𝑥 ∈ (𝐵 ↑_m 𝑆) ↦ (𝑥‘𝑋))
5	1, 2, 3, 4	mapsnf1o2 8821	. . 3 ⊢ (𝑥 ∈ (𝐵 ↑_m 𝑆) ↦ (𝑥‘𝑋)):(𝐵 ↑_m 𝑆)–1-1-onto→𝐵
6		f1ocnv 6776	. . 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 8820	. . . 4 ⊢ ^◡(𝑥 ∈ (𝐵 ↑_m 𝑆) ↦ (𝑥‘𝑋)) = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦}))
10	8, 9	eqtr4i 2755	. . 3 ⊢ 𝐹 = ^◡(𝑥 ∈ (𝐵 ↑_m 𝑆) ↦ (𝑥‘𝑋))
11		f1oeq1 6752	. . 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 1540 ∈ wcel 2109 Vcvv 3436 {csn 4577 ↦ cmpt 5173 × cxp 5617 ^◡ccnv 5618 –1-1-onto→wf1o 6481 ‘cfv 6482 (class class class)co 7349 ↑_m cmap 8753

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-ov 7352 df-oprab 7353 df-mpo 7354 df-1st 7924 df-2nd 7925 df-map 8755

This theorem is referenced by: coe1f2 22092 coe1add 22148 evls1rhmlem 22206 evl1sca 22219 pf1ind 22240 ismrer1 37838

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