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Mirrors > Home > MPE Home > Th. List > elrnmpt1s | Structured version Visualization version GIF version |
Description: Elementhood in an image set. (Contributed by Mario Carneiro, 12-Sep-2015.) |
Ref | Expression |
---|---|
rnmpt.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
elrnmpt1s.1 | ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
elrnmpt1s | ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . 3 ⊢ 𝐶 = 𝐶 | |
2 | elrnmpt1s.1 | . . . 4 ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶) | |
3 | 2 | rspceeqv 3586 | . . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 = 𝐶) → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵) |
4 | 1, 3 | mpan2 690 | . 2 ⊢ (𝐷 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵) |
5 | rnmpt.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
6 | 5 | elrnmpt 5792 | . . 3 ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)) |
7 | 6 | biimparc 483 | . 2 ⊢ ((∃𝑥 ∈ 𝐴 𝐶 = 𝐵 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ ran 𝐹) |
8 | 4, 7 | sylan 583 | 1 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 ↦ cmpt 5110 ran crn 5520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-mpt 5111 df-cnv 5527 df-dm 5529 df-rn 5530 |
This theorem is referenced by: wunex2 10149 dfod2 18683 dprd2dlem1 19156 dprd2da 19157 ordtbaslem 21793 subgntr 22712 opnsubg 22713 tgpconncomp 22718 tsmsxplem1 22758 xrge0gsumle 23438 xrge0tsms 23439 minveclem3b 24032 minveclem3 24033 minveclem4 24036 efsubm 25143 dchrisum0fno1 26095 fnpreimac 30434 xrge0tsmsd 30742 esumcvg 31455 esum2d 31462 msubco 32891 suprubrnmpt2 41890 infxrlbrnmpt2 42047 sge0xaddlem1 43072 |
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