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Mirrors > Home > ILE Home > Th. List > cnvimass | GIF version |
Description: A preimage under any class is included in the domain of the class. (Contributed by FL, 29-Jan-2007.) |
Ref | Expression |
---|---|
cnvimass | ⊢ (◡𝐴 “ 𝐵) ⊆ dom 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imassrn 4939 | . 2 ⊢ (◡𝐴 “ 𝐵) ⊆ ran ◡𝐴 | |
2 | dfdm4 4778 | . 2 ⊢ dom 𝐴 = ran ◡𝐴 | |
3 | 1, 2 | sseqtrri 3163 | 1 ⊢ (◡𝐴 “ 𝐵) ⊆ dom 𝐴 |
Colors of variables: wff set class |
Syntax hints: ⊆ wss 3102 ◡ccnv 4585 dom cdm 4586 ran crn 4587 “ cima 4589 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-br 3966 df-opab 4026 df-xp 4592 df-cnv 4594 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 |
This theorem is referenced by: fvimacnvi 5581 elpreima 5586 fconst4m 5687 nn0supp 9142 fisumss 11289 fprodssdc 11487 cnpnei 12619 cnclima 12623 cnntri 12624 cnntr 12625 cncnp 12630 cnrest2 12636 cndis 12641 txcnmpt 12673 txdis1cn 12678 hmeoimaf1o 12714 xmeter 12836 |
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