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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfaimafn | Structured version Visualization version GIF version |
Description: Alternate definition of the image of a function, analogous to dfimafn 6596. (Contributed by Alexander van der Vekens, 25-May-2017.) |
Ref | Expression |
---|---|
dfaimafn | ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝑦}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3883 | . . . . . 6 ⊢ (𝐴 ⊆ dom 𝐹 → (𝑥 ∈ 𝐴 → 𝑥 ∈ dom 𝐹)) | |
2 | funbrafvb 42871 | . . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹'''𝑥) = 𝑦 ↔ 𝑥𝐹𝑦)) | |
3 | 2 | ex 413 | . . . . . 6 ⊢ (Fun 𝐹 → (𝑥 ∈ dom 𝐹 → ((𝐹'''𝑥) = 𝑦 ↔ 𝑥𝐹𝑦))) |
4 | 1, 3 | syl9r 78 | . . . . 5 ⊢ (Fun 𝐹 → (𝐴 ⊆ dom 𝐹 → (𝑥 ∈ 𝐴 → ((𝐹'''𝑥) = 𝑦 ↔ 𝑥𝐹𝑦)))) |
5 | 4 | imp31 418 | . . . 4 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ 𝑥 ∈ 𝐴) → ((𝐹'''𝑥) = 𝑦 ↔ 𝑥𝐹𝑦)) |
6 | 5 | rexbidva 3259 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝑦 ↔ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦)) |
7 | 6 | abbidv 2860 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦}) |
8 | dfima2 5808 | . 2 ⊢ (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦} | |
9 | 7, 8 | syl6reqr 2850 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝑦}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1522 ∈ wcel 2081 {cab 2775 ∃wrex 3106 ⊆ wss 3859 class class class wbr 4962 dom cdm 5443 “ cima 5446 Fun wfun 6219 '''cafv 42832 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-int 4783 df-br 4963 df-opab 5025 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-iota 6189 df-fun 6227 df-fn 6228 df-fv 6233 df-aiota 42801 df-dfat 42834 df-afv 42835 |
This theorem is referenced by: dfaimafn2 42881 |
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