f1mptrn - Mathbox for Thierry Arnoux

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Theorem f1mptrn 32645

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 3146	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶)
3		f1mptrn.2	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ∃!𝑥 ∈ 𝐴 𝑦 = 𝐵)
4	3	ralrimiva 3146	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐶 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐵)
5		eqid 2737	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
6	5	f1ompt 7131	. . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴–1-1-onto→𝐶 ↔ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ ∀𝑦 ∈ 𝐶 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐵))
7		dff1o2 6853	. . . 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 1147	. . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴–1-1-onto→𝐶 → Fun ^◡(𝑥 ∈ 𝐴 ↦ 𝐵))
9	6, 8	sylbir 235	. 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 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃!wreu 3378 ↦ cmpt 5225 ^◡ccnv 5684 ran crn 5686 Fun wfun 6555 Fn wfn 6556 –1-1-onto→wf1o 6560

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568

This theorem is referenced by: esum2dlem 34093

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