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Mirrors > Home > MPE Home > Th. List > elrnmptg | Structured version Visualization version GIF version |
Description: Membership in the range of a function. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
rnmpt.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
Ref | Expression |
---|---|
elrnmptg | ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnmpt.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | 1 | rnmpt 5971 | . . 3 ⊢ ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} |
3 | 2 | eleq2i 2831 | . 2 ⊢ (𝐶 ∈ ran 𝐹 ↔ 𝐶 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) |
4 | r19.29 3112 | . . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵) → ∃𝑥 ∈ 𝐴 (𝐵 ∈ 𝑉 ∧ 𝐶 = 𝐵)) | |
5 | eleq1 2827 | . . . . . . . 8 ⊢ (𝐶 = 𝐵 → (𝐶 ∈ 𝑉 ↔ 𝐵 ∈ 𝑉)) | |
6 | 5 | biimparc 479 | . . . . . . 7 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 = 𝐵) → 𝐶 ∈ 𝑉) |
7 | 6 | elexd 3502 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 = 𝐵) → 𝐶 ∈ V) |
8 | 7 | rexlimivw 3149 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 (𝐵 ∈ 𝑉 ∧ 𝐶 = 𝐵) → 𝐶 ∈ V) |
9 | 4, 8 | syl 17 | . . . 4 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵) → 𝐶 ∈ V) |
10 | 9 | ex 412 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∃𝑥 ∈ 𝐴 𝐶 = 𝐵 → 𝐶 ∈ V)) |
11 | eqeq1 2739 | . . . . 5 ⊢ (𝑦 = 𝐶 → (𝑦 = 𝐵 ↔ 𝐶 = 𝐵)) | |
12 | 11 | rexbidv 3177 | . . . 4 ⊢ (𝑦 = 𝐶 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)) |
13 | 12 | elab3g 3688 | . . 3 ⊢ ((∃𝑥 ∈ 𝐴 𝐶 = 𝐵 → 𝐶 ∈ V) → (𝐶 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)) |
14 | 10, 13 | syl 17 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝐶 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)) |
15 | 3, 14 | bitrid 283 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {cab 2712 ∀wral 3059 ∃wrex 3068 Vcvv 3478 ↦ cmpt 5231 ran crn 5690 |
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-11 2155 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-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-mpt 5232 df-cnv 5697 df-dm 5699 df-rn 5700 |
This theorem is referenced by: elrnmpti 5976 iunrnmptss 32586 |
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