f1mptrn - Mathbox for Thierry Arnoux

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Theorem f1mptrn 32114

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 2732	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
6	5	f1ompt 7112	. . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴–1-1-onto→𝐶 ↔ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ ∀𝑦 ∈ 𝐶 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐵))
7		dff1o2 6838	. . . 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 1146	. . 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 1541 ∈ wcel 2106 ∀wral 3061 ∃!wreu 3374 ↦ cmpt 5231 ^◡ccnv 5675 ran crn 5677 Fun wfun 6537 Fn wfn 6538 –1-1-onto→wf1o 6542

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550

This theorem is referenced by: esum2dlem 33376

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