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Mirrors > Home > MPE Home > Th. List > f1imaen2g | Structured version Visualization version GIF version |
Description: If a function is one-to-one, then the image of a subset of its domain under it is equinumerous to the subset. (This version of f1imaeng 9012 does not need ax-rep 5278.) (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 25-Jun-2015.) |
Ref | Expression |
---|---|
f1imaen2g | ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → (𝐹 “ 𝐶) ≈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprr 770 | . . 3 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → 𝐶 ∈ 𝑉) | |
2 | simplr 766 | . . . 4 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → 𝐵 ∈ 𝑉) | |
3 | f1f 6781 | . . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵) | |
4 | fimass 6732 | . . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝐶) ⊆ 𝐵) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐹 “ 𝐶) ⊆ 𝐵) |
6 | 5 | ad2antrr 723 | . . . 4 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → (𝐹 “ 𝐶) ⊆ 𝐵) |
7 | 2, 6 | ssexd 5317 | . . 3 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → (𝐹 “ 𝐶) ∈ V) |
8 | f1ores 6841 | . . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶)) | |
9 | 8 | ad2ant2r 744 | . . 3 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶)) |
10 | f1oen2g 8966 | . . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ (𝐹 “ 𝐶) ∈ V ∧ (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶)) → 𝐶 ≈ (𝐹 “ 𝐶)) | |
11 | 1, 7, 9, 10 | syl3anc 1368 | . 2 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → 𝐶 ≈ (𝐹 “ 𝐶)) |
12 | 11 | ensymd 9003 | 1 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → (𝐹 “ 𝐶) ≈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2098 Vcvv 3468 ⊆ wss 3943 class class class wbr 5141 ↾ cres 5671 “ cima 5672 ⟶wf 6533 –1-1→wf1 6534 –1-1-onto→wf1o 6536 ≈ cen 8938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-er 8705 df-en 8942 |
This theorem is referenced by: ssenen 9153 phplem4OLD 9222 fiint 9326 unxpwdom2 9585 znunithash 21459 |
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