mapsnf1o2 - Metamath Proof Explorer

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Theorem mapsnf1o2 8461

Description: Explicit bijection between a set and its singleton functions. (Contributed by Stefan O'Rear, 21-Mar-2015.)

**Hypotheses**
Ref	Expression
mapsncnv.s	⊢ 𝑆 = {𝑋}
mapsncnv.b	⊢ 𝐵 ∈ V
mapsncnv.x	⊢ 𝑋 ∈ V
mapsncnv.f	⊢ 𝐹 = (𝑥 ∈ (𝐵 ↑_m 𝑆) ↦ (𝑥‘𝑋))

**Assertion**
Ref	Expression
mapsnf1o2	⊢ 𝐹:(𝐵 ↑_m 𝑆)–1-1-onto→𝐵

Distinct variable groups: 𝑥,𝐵 𝑥,𝑆 Allowed substitution hints: 𝐹(𝑥) 𝑋(𝑥)

Proof of Theorem mapsnf1o2 Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6686	. . 3 ⊢ (𝑥‘𝑋) ∈ V
2		mapsncnv.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ (𝐵 ↑_m 𝑆) ↦ (𝑥‘𝑋))
3	1, 2	fnmpti 6494	. 2 ⊢ 𝐹 Fn (𝐵 ↑_m 𝑆)
4		mapsncnv.s	. . . . 5 ⊢ 𝑆 = {𝑋}
5		snex 5335	. . . . 5 ⊢ {𝑋} ∈ V
6	4, 5	eqeltri 2912	. . . 4 ⊢ 𝑆 ∈ V
7		snex 5335	. . . 4 ⊢ {𝑦} ∈ V
8	6, 7	xpex 7479	. . 3 ⊢ (𝑆 × {𝑦}) ∈ V
9		mapsncnv.b	. . . 4 ⊢ 𝐵 ∈ V
10		mapsncnv.x	. . . 4 ⊢ 𝑋 ∈ V
11	4, 9, 10, 2	mapsncnv 8460	. . 3 ⊢ ^◡𝐹 = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦}))
12	8, 11	fnmpti 6494	. 2 ⊢ ^◡𝐹 Fn 𝐵
13		dff1o4 6626	. 2 ⊢ (𝐹:(𝐵 ↑_m 𝑆)–1-1-onto→𝐵 ↔ (𝐹 Fn (𝐵 ↑_m 𝑆) ∧ ^◡𝐹 Fn 𝐵))
14	3, 12, 13	mpbir2an 709	1 ⊢ 𝐹:(𝐵 ↑_m 𝑆)–1-1-onto→𝐵

Colors of variables: wff setvar class

Syntax hints: = wceq 1536 ∈ wcel 2113 Vcvv 3497 {csn 4570 ↦ cmpt 5149 × cxp 5556 ^◡ccnv 5557 Fn wfn 6353 –1-1-onto→wf1o 6357 ‘cfv 6358 (class class class)co 7159 ↑_m cmap 8409

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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-1st 7692 df-2nd 7693 df-map 8411

This theorem is referenced by: mapsnf1o3 8462 coe1sfi 20384 coe1mul2lem2 20439 ply1coe 20467 evl1var 20502 pf1mpf 20518 pf1ind 20521 deg1ldg 24689 deg1leb 24692

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