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Mirrors > Home > MPE Home > Th. List > funeqi | Structured version Visualization version GIF version |
Description: Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
Ref | Expression |
---|---|
funeqi.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
funeqi | ⊢ (Fun 𝐴 ↔ Fun 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funeqi.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | funeq 6587 | . 2 ⊢ (𝐴 = 𝐵 → (Fun 𝐴 ↔ Fun 𝐵)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (Fun 𝐴 ↔ Fun 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1536 Fun wfun 6556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ss 3979 df-br 5148 df-opab 5210 df-rel 5695 df-cnv 5696 df-co 5697 df-fun 6564 |
This theorem is referenced by: funmpt 6605 funmpt2 6606 funco 6607 funresfunco 6608 fununfun 6615 funprg 6621 funtpg 6622 funtp 6624 funcnvpr 6629 funcnvtp 6630 funcnvqp 6631 funcnv0 6633 f1cnvcnv 6813 f1cof1 6814 opabiotafun 6988 fvn0ssdmfun 7093 funopdmsn 7169 fpropnf1 7286 funoprabg 7553 mpofun 7556 ovidig 7574 funcnvuni 7954 fiun 7965 f1iun 7966 tposfun 8265 tfr1a 8432 tz7.44lem1 8443 tz7.48-2 8480 ssdomg 9038 sbthlem7 9127 sbthlem8 9128 1sdomOLD 9282 hartogslem1 9579 r1funlim 9803 zorn2lem4 10536 axaddf 11182 axmulf 11183 fundmge2nop0 14537 funcnvs1 14947 strleun 17190 fthoppc 17976 cnfldfun 21395 cnfldfunALT 21396 cnfldfunOLD 21408 cnfldfunALTOLD 21409 cnfldfunALTOLDOLD 21410 volf 25577 dfrelog 26621 precsexlem10 28254 precsexlem11 28255 usgredg3 29247 ushgredgedg 29260 ushgredgedgloop 29262 2trld 29967 0pth 30153 1pthdlem1 30163 1trld 30170 3trld 30200 ajfuni 30887 hlimf 31265 funadj 31914 funcnvadj 31921 rinvf1o 32646 bnj97 34858 bnj150 34868 bnj1384 35024 bnj1421 35034 bnj60 35054 satffunlem2lem2 35390 satfv0fvfmla0 35397 funpartfun 35924 funtransport 36012 funray 36121 funline 36123 modelaxreplem2 44943 xlimfun 45810 funcoressn 46991 |
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