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Theorem oacomf1olem 8591
Description: Lemma for oacomf1o 8592. (Contributed by Mario Carneiro, 30-May-2015.)
Hypothesis
Ref Expression
oacomf1olem.1 𝐹 = (π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯))
Assertion
Ref Expression
oacomf1olem ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (𝐹:𝐴–1-1-ontoβ†’ran 𝐹 ∧ (ran 𝐹 ∩ 𝐡) = βˆ…))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝐡
Allowed substitution hint:   𝐹(π‘₯)

Proof of Theorem oacomf1olem
StepHypRef Expression
1 oaf1o 8590 . . . . . . 7 (𝐡 ∈ On β†’ (π‘₯ ∈ On ↦ (𝐡 +o π‘₯)):On–1-1-ontoβ†’(On βˆ– 𝐡))
21adantl 480 . . . . . 6 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (π‘₯ ∈ On ↦ (𝐡 +o π‘₯)):On–1-1-ontoβ†’(On βˆ– 𝐡))
3 f1of1 6843 . . . . . 6 ((π‘₯ ∈ On ↦ (𝐡 +o π‘₯)):On–1-1-ontoβ†’(On βˆ– 𝐡) β†’ (π‘₯ ∈ On ↦ (𝐡 +o π‘₯)):On–1-1β†’(On βˆ– 𝐡))
42, 3syl 17 . . . . 5 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (π‘₯ ∈ On ↦ (𝐡 +o π‘₯)):On–1-1β†’(On βˆ– 𝐡))
5 onss 7793 . . . . . 6 (𝐴 ∈ On β†’ 𝐴 βŠ† On)
65adantr 479 . . . . 5 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ 𝐴 βŠ† On)
7 f1ssres 6806 . . . . 5 (((π‘₯ ∈ On ↦ (𝐡 +o π‘₯)):On–1-1β†’(On βˆ– 𝐡) ∧ 𝐴 βŠ† On) β†’ ((π‘₯ ∈ On ↦ (𝐡 +o π‘₯)) β†Ύ 𝐴):𝐴–1-1β†’(On βˆ– 𝐡))
84, 6, 7syl2anc 582 . . . 4 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ ((π‘₯ ∈ On ↦ (𝐡 +o π‘₯)) β†Ύ 𝐴):𝐴–1-1β†’(On βˆ– 𝐡))
96resmptd 6049 . . . . . 6 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ ((π‘₯ ∈ On ↦ (𝐡 +o π‘₯)) β†Ύ 𝐴) = (π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯)))
10 oacomf1olem.1 . . . . . 6 𝐹 = (π‘₯ ∈ 𝐴 ↦ (𝐡 +o π‘₯))
119, 10eqtr4di 2786 . . . . 5 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ ((π‘₯ ∈ On ↦ (𝐡 +o π‘₯)) β†Ύ 𝐴) = 𝐹)
12 f1eq1 6793 . . . . 5 (((π‘₯ ∈ On ↦ (𝐡 +o π‘₯)) β†Ύ 𝐴) = 𝐹 β†’ (((π‘₯ ∈ On ↦ (𝐡 +o π‘₯)) β†Ύ 𝐴):𝐴–1-1β†’(On βˆ– 𝐡) ↔ 𝐹:𝐴–1-1β†’(On βˆ– 𝐡)))
1311, 12syl 17 . . . 4 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (((π‘₯ ∈ On ↦ (𝐡 +o π‘₯)) β†Ύ 𝐴):𝐴–1-1β†’(On βˆ– 𝐡) ↔ 𝐹:𝐴–1-1β†’(On βˆ– 𝐡)))
148, 13mpbid 231 . . 3 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ 𝐹:𝐴–1-1β†’(On βˆ– 𝐡))
15 f1f1orn 6855 . . 3 (𝐹:𝐴–1-1β†’(On βˆ– 𝐡) β†’ 𝐹:𝐴–1-1-ontoβ†’ran 𝐹)
1614, 15syl 17 . 2 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ 𝐹:𝐴–1-1-ontoβ†’ran 𝐹)
17 f1f 6798 . . . 4 (𝐹:𝐴–1-1β†’(On βˆ– 𝐡) β†’ 𝐹:𝐴⟢(On βˆ– 𝐡))
18 frn 6734 . . . 4 (𝐹:𝐴⟢(On βˆ– 𝐡) β†’ ran 𝐹 βŠ† (On βˆ– 𝐡))
1914, 17, 183syl 18 . . 3 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ ran 𝐹 βŠ† (On βˆ– 𝐡))
2019difss2d 4135 . . . 4 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ ran 𝐹 βŠ† On)
21 reldisj 4455 . . . 4 (ran 𝐹 βŠ† On β†’ ((ran 𝐹 ∩ 𝐡) = βˆ… ↔ ran 𝐹 βŠ† (On βˆ– 𝐡)))
2220, 21syl 17 . . 3 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ ((ran 𝐹 ∩ 𝐡) = βˆ… ↔ ran 𝐹 βŠ† (On βˆ– 𝐡)))
2319, 22mpbird 256 . 2 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (ran 𝐹 ∩ 𝐡) = βˆ…)
2416, 23jca 510 1 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (𝐹:𝐴–1-1-ontoβ†’ran 𝐹 ∧ (ran 𝐹 ∩ 𝐡) = βˆ…))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098   βˆ– cdif 3946   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4326   ↦ cmpt 5235  ran crn 5683   β†Ύ cres 5684  Oncon0 6374  βŸΆwf 6549  β€“1-1β†’wf1 6550  β€“1-1-ontoβ†’wf1o 6552  (class class class)co 7426   +o coa 8490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-oadd 8497
This theorem is referenced by:  oacomf1o  8592  onadju  10224
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