f1mptrn - Mathbox for Thierry Arnoux

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Theorem f1mptrn 32961

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 3157	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶)
3		f1mptrn.2	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ∃!𝑥 ∈ 𝐴 𝑦 = 𝐵)
4	3	ralrimiva 3157	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐶 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐵)
5		eqid 2763	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
6	5	f1ompt 7108	. . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴–1-1-onto→𝐶 ↔ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ ∀𝑦 ∈ 𝐶 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐵))
7		dff1o2 6828	. . . 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 1164	. . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴–1-1-onto→𝐶 → Fun ^◡(𝑥 ∈ 𝐴 ↦ 𝐵))
9	6, 8	sylbir 238	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ ∀𝑦 ∈ 𝐶 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐵) → Fun ^◡(𝑥 ∈ 𝐴 ↦ 𝐵))
10	2, 4, 9	syl2anc 595	1 ⊢ (𝜑 → Fun ^◡(𝑥 ∈ 𝐴 ↦ 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ∀wral 3079 ∃!wreu 3367 ↦ cmpt 5193 ^◡ccnv 5662 ran crn 5664 Fun wfun 6532 Fn wfn 6533 –1-1-onto→wf1o 6537

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5258 ax-pr 5406

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-if 4489 df-sn 4591 df-pr 4593 df-op 4597 df-br 5111 df-opab 5175 df-mpt 5194 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545

This theorem is referenced by: esum2dlem 34463

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