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Theorem dff4im 5227
Description: Property of a mapping. (Contributed by Jim Kingdon, 4-Jan-2019.)
Assertion
Ref Expression
dff4im (𝐹:AB → (𝐹 ⊆ (A × B) x A ∃!y B x𝐹y))
Distinct variable groups:   x,y,A   x,B,y   x,𝐹,y

Proof of Theorem dff4im
StepHypRef Expression
1 dff3im 5226 . 2 (𝐹:AB → (𝐹 ⊆ (A × B) x A ∃!y x𝐹y))
2 df-br 3729 . . . . . . . 8 (x𝐹y ↔ ⟨x, y 𝐹)
3 ssel 2908 . . . . . . . . 9 (𝐹 ⊆ (A × B) → (⟨x, y 𝐹 → ⟨x, y (A × B)))
4 opelxp2 4294 . . . . . . . . 9 (⟨x, y (A × B) → y B)
53, 4syl6 29 . . . . . . . 8 (𝐹 ⊆ (A × B) → (⟨x, y 𝐹y B))
62, 5syl5bi 141 . . . . . . 7 (𝐹 ⊆ (A × B) → (x𝐹yy B))
76pm4.71rd 374 . . . . . 6 (𝐹 ⊆ (A × B) → (x𝐹y ↔ (y B x𝐹y)))
87eubidv 1882 . . . . 5 (𝐹 ⊆ (A × B) → (∃!y x𝐹y∃!y(y B x𝐹y)))
9 df-reu 2283 . . . . 5 (∃!y B x𝐹y∃!y(y B x𝐹y))
108, 9syl6bbr 187 . . . 4 (𝐹 ⊆ (A × B) → (∃!y x𝐹y∃!y B x𝐹y))
1110ralbidv 2296 . . 3 (𝐹 ⊆ (A × B) → (x A ∃!y x𝐹yx A ∃!y B x𝐹y))
1211pm5.32i 427 . 2 ((𝐹 ⊆ (A × B) x A ∃!y x𝐹y) ↔ (𝐹 ⊆ (A × B) x A ∃!y B x𝐹y))
131, 12sylib 127 1 (𝐹:AB → (𝐹 ⊆ (A × B) x A ∃!y B x𝐹y))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1367  ∃!weu 1874  wral 2276  ∃!wreu 2278  wss 2886  cop 3343   class class class wbr 3728   × cxp 4259  wf 4814
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 614  ax-5 1310  ax-7 1311  ax-gen 1312  ax-ie1 1356  ax-ie2 1357  ax-8 1369  ax-10 1370  ax-11 1371  ax-i12 1372  ax-bnd 1373  ax-4 1374  ax-14 1379  ax-17 1393  ax-i9 1397  ax-ial 1401  ax-i5r 1402  ax-ext 1996  ax-sep 3839  ax-pow 3891  ax-pr 3908
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1227  df-nf 1324  df-sb 1620  df-eu 1877  df-mo 1878  df-clab 2001  df-cleq 2007  df-clel 2010  df-nfc 2141  df-ral 2281  df-rex 2282  df-reu 2283  df-v 2529  df-sbc 2734  df-un 2891  df-in 2893  df-ss 2900  df-pw 3326  df-sn 3346  df-pr 3347  df-op 3349  df-uni 3545  df-br 3729  df-opab 3783  df-id 3994  df-xp 4267  df-rel 4268  df-cnv 4269  df-co 4270  df-dm 4271  df-rn 4272  df-iota 4783  df-fun 4820  df-fn 4821  df-f 4822  df-fv 4826
This theorem is referenced by: (None)
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