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Mirrors > Home > ILE Home > Th. List > f1imaen2g | GIF version |
Description: A one-to-one function's image under a subset of its domain is equinumerous to the subset. (This version of f1imaen 6681 does not need ax-setind 4447.) (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 521 | . . 3 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → 𝐶 ∈ 𝑉) | |
2 | simplr 519 | . . . 4 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → 𝐵 ∈ 𝑉) | |
3 | f1f 5323 | . . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵) | |
4 | imassrn 4887 | . . . . . . 7 ⊢ (𝐹 “ 𝐶) ⊆ ran 𝐹 | |
5 | frn 5276 | . . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵) | |
6 | 4, 5 | sstrid 3103 | . . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝐶) ⊆ 𝐵) |
7 | 3, 6 | syl 14 | . . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐹 “ 𝐶) ⊆ 𝐵) |
8 | 7 | ad2antrr 479 | . . . 4 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → (𝐹 “ 𝐶) ⊆ 𝐵) |
9 | 2, 8 | ssexd 4063 | . . 3 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → (𝐹 “ 𝐶) ∈ V) |
10 | f1ores 5375 | . . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶)) | |
11 | 10 | ad2ant2r 500 | . . 3 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶)) |
12 | f1oen2g 6642 | . . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ (𝐹 “ 𝐶) ∈ V ∧ (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶)) → 𝐶 ≈ (𝐹 “ 𝐶)) | |
13 | 1, 9, 11, 12 | syl3anc 1216 | . 2 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → 𝐶 ≈ (𝐹 “ 𝐶)) |
14 | 13 | ensymd 6670 | 1 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → (𝐹 “ 𝐶) ≈ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1480 Vcvv 2681 ⊆ wss 3066 class class class wbr 3924 ran crn 4535 ↾ cres 4536 “ cima 4537 ⟶wf 5114 –1-1→wf1 5115 –1-1-onto→wf1o 5117 ≈ cen 6625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-er 6422 df-en 6628 |
This theorem is referenced by: ssenen 6738 phplem4 6742 phplem4dom 6749 phplem4on 6754 fiintim 6810 |
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