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Mirrors > Home > ILE Home > Th. List > elrnrexdm | Unicode version |
Description: For any element in the range of a function there is an element in the domain of the function for which the function value is the element of the range. (Contributed by Alexander van der Vekens, 8-Dec-2017.) |
Ref | Expression |
---|---|
elrnrexdm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2171 | . . . . . 6 | |
2 | 1 | ancli 321 | . . . . 5 |
3 | 2 | adantl 275 | . . . 4 |
4 | eqeq2 2180 | . . . . 5 | |
5 | 4 | rspcev 2834 | . . . 4 |
6 | 3, 5 | syl 14 | . . 3 |
7 | 6 | ex 114 | . 2 |
8 | funfn 5228 | . . 3 | |
9 | eqeq2 2180 | . . . 4 | |
10 | 9 | rexrn 5633 | . . 3 |
11 | 8, 10 | sylbi 120 | . 2 |
12 | 7, 11 | sylibd 148 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1348 wcel 2141 wrex 2449 cdm 4611 crn 4612 wfun 5192 wfn 5193 cfv 5198 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-iota 5160 df-fun 5200 df-fn 5201 df-fv 5206 |
This theorem is referenced by: cc2lem 7228 ennnfonelemrnh 12371 ennnfonelemf1 12373 exmidsbthrlem 14054 |
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