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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > funimassd | Structured version Visualization version GIF version |
Description: Sufficient condition for the image of a function being a subclass. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
funimassd.1 | ⊢ Ⅎ𝑥𝜑 |
funimassd.2 | ⊢ (𝜑 → Fun 𝐹) |
funimassd.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵) |
Ref | Expression |
---|---|
funimassd | ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funimassd.2 | . . . 4 ⊢ (𝜑 → Fun 𝐹) | |
2 | fvelima 6706 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ (𝐹 “ 𝐴)) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦) | |
3 | 1, 2 | sylan 583 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ 𝐴)) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦) |
4 | funimassd.1 | . . . . 5 ⊢ Ⅎ𝑥𝜑 | |
5 | nfv 1915 | . . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ (𝐹 “ 𝐴) | |
6 | 4, 5 | nfan 1900 | . . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ (𝐹 “ 𝐴)) |
7 | nfv 1915 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵 | |
8 | id 22 | . . . . . . . . 9 ⊢ ((𝐹‘𝑥) = 𝑦 → (𝐹‘𝑥) = 𝑦) | |
9 | 8 | eqcomd 2804 | . . . . . . . 8 ⊢ ((𝐹‘𝑥) = 𝑦 → 𝑦 = (𝐹‘𝑥)) |
10 | 9 | 3ad2ant3 1132 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦) → 𝑦 = (𝐹‘𝑥)) |
11 | funimassd.3 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵) | |
12 | 11 | 3adant3 1129 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦) → (𝐹‘𝑥) ∈ 𝐵) |
13 | 10, 12 | eqeltrd 2890 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦) → 𝑦 ∈ 𝐵) |
14 | 13 | 3exp 1116 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐵))) |
15 | 14 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ 𝐴)) → (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐵))) |
16 | 6, 7, 15 | rexlimd 3276 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ 𝐴)) → (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐵)) |
17 | 3, 16 | mpd 15 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ 𝐴)) → 𝑦 ∈ 𝐵) |
18 | 17 | ssd 41716 | 1 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 ∃wrex 3107 ⊆ wss 3881 “ cima 5522 Fun wfun 6318 ‘cfv 6324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fv 6332 |
This theorem is referenced by: funimaeq 41884 |
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