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Theorem f1oun 5304
 Description: The union of two one-to-one onto functions with disjoint domains and ranges. (Contributed by set.mm contributors, 26-Mar-1998.)
Assertion
Ref Expression
f1oun (((F:A1-1-ontoB G:C1-1-ontoD) ((AC) = (BD) = )) → (FG):(AC)–1-1-onto→(BD))

Proof of Theorem f1oun
StepHypRef Expression
1 dff1o4 5294 . . . 4 (F:A1-1-ontoB ↔ (F Fn A F Fn B))
2 dff1o4 5294 . . . 4 (G:C1-1-ontoD ↔ (G Fn C G Fn D))
3 fnun 5189 . . . . . . 7 (((F Fn A G Fn C) (AC) = ) → (FG) Fn (AC))
43ex 423 . . . . . 6 ((F Fn A G Fn C) → ((AC) = → (FG) Fn (AC)))
5 fnun 5189 . . . . . . . 8 (((F Fn B G Fn D) (BD) = ) → (FG) Fn (BD))
6 cnvun 5033 . . . . . . . . 9 (FG) = (FG)
76fneq1i 5178 . . . . . . . 8 ((FG) Fn (BD) ↔ (FG) Fn (BD))
85, 7sylibr 203 . . . . . . 7 (((F Fn B G Fn D) (BD) = ) → (FG) Fn (BD))
98ex 423 . . . . . 6 ((F Fn B G Fn D) → ((BD) = (FG) Fn (BD)))
104, 9im2anan9 808 . . . . 5 (((F Fn A G Fn C) (F Fn B G Fn D)) → (((AC) = (BD) = ) → ((FG) Fn (AC) (FG) Fn (BD))))
1110an4s 799 . . . 4 (((F Fn A F Fn B) (G Fn C G Fn D)) → (((AC) = (BD) = ) → ((FG) Fn (AC) (FG) Fn (BD))))
121, 2, 11syl2anb 465 . . 3 ((F:A1-1-ontoB G:C1-1-ontoD) → (((AC) = (BD) = ) → ((FG) Fn (AC) (FG) Fn (BD))))
13 dff1o4 5294 . . 3 ((FG):(AC)–1-1-onto→(BD) ↔ ((FG) Fn (AC) (FG) Fn (BD)))
1412, 13syl6ibr 218 . 2 ((F:A1-1-ontoB G:C1-1-ontoD) → (((AC) = (BD) = ) → (FG):(AC)–1-1-onto→(BD)))
1514imp 418 1 (((F:A1-1-ontoB G:C1-1-ontoD) ((AC) = (BD) = )) → (FG):(AC)–1-1-onto→(BD))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∪ cun 3207   ∩ cin 3208  ∅c0 3550  ◡ccnv 4771   Fn wfn 4776  –1-1-onto→wf1o 4780 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794 This theorem is referenced by:  unen  6048
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