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Mirrors > Home > MPE Home > Th. List > 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 | . . . . 5 ⊢ (𝜑 → Fun 𝐹) | |
2 | fvelima 6974 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ (𝐹 “ 𝐴)) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦) | |
3 | 1, 2 | sylan 580 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ 𝐴)) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦) |
4 | funimassd.1 | . . . . . 6 ⊢ Ⅎ𝑥𝜑 | |
5 | nfv 1912 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ (𝐹 “ 𝐴) | |
6 | 4, 5 | nfan 1897 | . . . . 5 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ (𝐹 “ 𝐴)) |
7 | nfv 1912 | . . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵 | |
8 | id 22 | . . . . . . . . . 10 ⊢ ((𝐹‘𝑥) = 𝑦 → (𝐹‘𝑥) = 𝑦) | |
9 | 8 | eqcomd 2741 | . . . . . . . . 9 ⊢ ((𝐹‘𝑥) = 𝑦 → 𝑦 = (𝐹‘𝑥)) |
10 | 9 | 3ad2ant3 1134 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦) → 𝑦 = (𝐹‘𝑥)) |
11 | funimassd.3 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵) | |
12 | 11 | 3adant3 1131 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦) → (𝐹‘𝑥) ∈ 𝐵) |
13 | 10, 12 | eqeltrd 2839 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦) → 𝑦 ∈ 𝐵) |
14 | 13 | 3exp 1118 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐵))) |
15 | 14 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ 𝐴)) → (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐵))) |
16 | 6, 7, 15 | rexlimd 3264 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ 𝐴)) → (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐵)) |
17 | 3, 16 | mpd 15 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ 𝐴)) → 𝑦 ∈ 𝐵) |
18 | 17 | ex 412 | . 2 ⊢ (𝜑 → (𝑦 ∈ (𝐹 “ 𝐴) → 𝑦 ∈ 𝐵)) |
19 | 18 | ssrdv 4001 | 1 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 Ⅎwnf 1780 ∈ wcel 2106 ∃wrex 3068 ⊆ wss 3963 “ cima 5692 Fun wfun 6557 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fv 6571 |
This theorem is referenced by: ig1pmindeg 33602 aks6d1c3 42105 aks6d1c2lem4 42109 aks6d1c2 42112 aks6d1c6lem2 42153 funimaeq 45191 |
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