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Mirrors > Home > MPE Home > Th. List > Mathboxes > f1mptrn | Structured version Visualization version GIF version |
Description: Express injection for a mapping operation. (Contributed by Thierry Arnoux, 3-May-2020.) |
Ref | Expression |
---|---|
f1mptrn.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) |
f1mptrn.2 | ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ∃!𝑥 ∈ 𝐴 𝑦 = 𝐵) |
Ref | Expression |
---|---|
f1mptrn | ⊢ (𝜑 → Fun ◡(𝑥 ∈ 𝐴 ↦ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1mptrn.1 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) | |
2 | 1 | ralrimiva 3182 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) |
3 | f1mptrn.2 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ∃!𝑥 ∈ 𝐴 𝑦 = 𝐵) | |
4 | 3 | ralrimiva 3182 | . 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐶 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐵) |
5 | eqid 2821 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
6 | 5 | f1ompt 6870 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴–1-1-onto→𝐶 ↔ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ ∀𝑦 ∈ 𝐶 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐵)) |
7 | dff1o2 6615 | . . . 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 1142 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴–1-1-onto→𝐶 → Fun ◡(𝑥 ∈ 𝐴 ↦ 𝐵)) |
9 | 6, 8 | sylbir 237 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ ∀𝑦 ∈ 𝐶 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐵) → Fun ◡(𝑥 ∈ 𝐴 ↦ 𝐵)) |
10 | 2, 4, 9 | syl2anc 586 | 1 ⊢ (𝜑 → Fun ◡(𝑥 ∈ 𝐴 ↦ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∃!wreu 3140 ↦ cmpt 5139 ◡ccnv 5549 ran crn 5551 Fun wfun 6344 Fn wfn 6345 –1-1-onto→wf1o 6349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 |
This theorem is referenced by: esum2dlem 31346 |
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