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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 6724 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ (𝐹 “ 𝐴)) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦) | |
3 | 1, 2 | sylan 580 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ 𝐴)) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦) |
4 | funimassd.1 | . . . . 5 ⊢ Ⅎ𝑥𝜑 | |
5 | nfv 1906 | . . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ (𝐹 “ 𝐴) | |
6 | 4, 5 | nfan 1891 | . . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ (𝐹 “ 𝐴)) |
7 | nfv 1906 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵 | |
8 | id 22 | . . . . . . . . 9 ⊢ ((𝐹‘𝑥) = 𝑦 → (𝐹‘𝑥) = 𝑦) | |
9 | 8 | eqcomd 2824 | . . . . . . . 8 ⊢ ((𝐹‘𝑥) = 𝑦 → 𝑦 = (𝐹‘𝑥)) |
10 | 9 | 3ad2ant3 1127 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦) → 𝑦 = (𝐹‘𝑥)) |
11 | funimassd.3 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵) | |
12 | 11 | 3adant3 1124 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦) → (𝐹‘𝑥) ∈ 𝐵) |
13 | 10, 12 | eqeltrd 2910 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦) → 𝑦 ∈ 𝐵) |
14 | 13 | 3exp 1111 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐵))) |
15 | 14 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ 𝐴)) → (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐵))) |
16 | 6, 7, 15 | rexlimd 3314 | . . 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 41221 | 1 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 Ⅎwnf 1775 ∈ wcel 2105 ∃wrex 3136 ⊆ wss 3933 “ cima 5551 Fun wfun 6342 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fv 6356 |
This theorem is referenced by: funimaeq 41394 |
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