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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 1914 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ (𝐹 “ 𝐴) | |
| 6 | 4, 5 | nfan 1899 | . . . . 5 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ (𝐹 “ 𝐴)) |
| 7 | nfv 1914 | . . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵 | |
| 8 | id 22 | . . . . . . . . . 10 ⊢ ((𝐹‘𝑥) = 𝑦 → (𝐹‘𝑥) = 𝑦) | |
| 9 | 8 | eqcomd 2743 | . . . . . . . . 9 ⊢ ((𝐹‘𝑥) = 𝑦 → 𝑦 = (𝐹‘𝑥)) |
| 10 | 9 | 3ad2ant3 1136 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦) → 𝑦 = (𝐹‘𝑥)) |
| 11 | funimassd.3 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵) | |
| 12 | 11 | 3adant3 1133 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦) → (𝐹‘𝑥) ∈ 𝐵) |
| 13 | 10, 12 | eqeltrd 2841 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑦) → 𝑦 ∈ 𝐵) |
| 14 | 13 | 3exp 1120 | . . . . . 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 3266 | . . . 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 3989 | 1 ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 ∃wrex 3070 ⊆ wss 3951 “ cima 5688 Fun wfun 6555 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fv 6569 |
| This theorem is referenced by: ig1pmindeg 33622 exsslsb 33647 aks6d1c3 42124 aks6d1c2lem4 42128 aks6d1c2 42131 aks6d1c6lem2 42172 funimaeq 45253 |
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