Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) | |
2 | 1 | ralrimiva 2949 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) |
3 | f1mptrn.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ∃!𝑥 ∈ 𝐴 𝑦 = 𝐵) | |
4 | 3 | ralrimiva 2949 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐶 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐵) |
5 | 2, 4 | jca 553 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ ∀𝑦 ∈ 𝐶 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐵)) |
6 | eqid 2610 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
7 | 6 | f1ompt 6290 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴–1-1-onto→𝐶 ↔ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ ∀𝑦 ∈ 𝐶 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐵)) |
8 | dff1o2 6055 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴–1-1-onto→𝐶 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ Fun ◡(𝑥 ∈ 𝐴 ↦ 𝐵) ∧ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐶)) | |
9 | 8 | simp2bi 1070 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴–1-1-onto→𝐶 → Fun ◡(𝑥 ∈ 𝐴 ↦ 𝐵)) |
10 | 7, 9 | sylbir 224 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ ∀𝑦 ∈ 𝐶 ∃!𝑥 ∈ 𝐴 𝑦 = 𝐵) → Fun ◡(𝑥 ∈ 𝐴 ↦ 𝐵)) |
11 | 5, 10 | syl 17 | 1 ⊢ (𝜑 → Fun ◡(𝑥 ∈ 𝐴 ↦ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃!wreu 2898 ↦ cmpt 4643 ◡ccnv 5037 ran crn 5039 Fun wfun 5798 Fn wfn 5799 –1-1-onto→wf1o 5803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 |
This theorem is referenced by: esum2dlem 29481 |
Copyright terms: Public domain | W3C validator |