f1mptrn - Mathbox for Thierry Arnoux

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Theorem f1mptrn 31105

Description: Express injection for a mapping operation. (Contributed by Thierry Arnoux, 3-May-2020.)

**Hypotheses**
Ref	Expression
f1mptrn.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)
f1mptrn.2	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ∃!𝑥 ∈ 𝐴 𝑦 = 𝐵)

**Assertion**
Ref	Expression
f1mptrn	⊢ (𝜑 → Fun ^◡(𝑥 ∈ 𝐴 ↦ 𝐵))

Distinct variable groups: 𝑥,𝐴,𝑦 𝑦,𝐵 𝑥,𝐶,𝑦 𝜑,𝑥,𝑦 Allowed substitution hint: 𝐵(𝑥)

**Proof of Theorem f1mptrn**
Step	Hyp	Ref	Expression
1		f1mptrn.1	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)
2	1	ralrimiva 3140	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶)
3		f1mptrn.2	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ∃!𝑥 ∈ 𝐴 𝑦 = 𝐵)
4	3	ralrimiva 3140	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐶 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐵)
5		eqid 2737	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
6	5	f1ompt 7025	. . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴–1-1-onto→𝐶 ↔ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ ∀𝑦 ∈ 𝐶 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐵))
7		dff1o2 6759	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴–1-1-onto→𝐶 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ Fun ^◡(𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐶))
8	7	simp2bi 1145	. . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴–1-1-onto→𝐶 → Fun ^◡(𝑥 ∈ 𝐴 ↦ 𝐵))
9	6, 8	sylbir 234	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ ∀𝑦 ∈ 𝐶 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐵) → Fun ^◡(𝑥 ∈ 𝐴 ↦ 𝐵))
10	2, 4, 9	syl2anc 584	1 ⊢ (𝜑 → Fun ^◡(𝑥 ∈ 𝐴 ↦ 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1540 ∈ wcel 2105 ∀wral 3062 ∃!wreu 3348 ↦ cmpt 5170 ^◡ccnv 5607 ran crn 5609 Fun wfun 6460 Fn wfn 6461 –1-1-onto→wf1o 6465

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2708 ax-sep 5238 ax-nul 5245 ax-pr 5367

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ral 3063 df-rex 3072 df-reu 3351 df-rab 3405 df-v 3443 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4268 df-if 4472 df-sn 4572 df-pr 4574 df-op 4578 df-br 5088 df-opab 5150 df-mpt 5171 df-id 5507 df-xp 5614 df-rel 5615 df-cnv 5616 df-co 5617 df-dm 5618 df-rn 5619 df-res 5620 df-ima 5621 df-fun 6468 df-fn 6469 df-f 6470 df-f1 6471 df-fo 6472 df-f1o 6473

This theorem is referenced by: esum2dlem 32200

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