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Theorem oacomf1olem 8515
Description: Lemma for oacomf1o 8516. (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 8514 . . . . . . 7 (𝐡 ∈ On β†’ (π‘₯ ∈ On ↦ (𝐡 +o π‘₯)):On–1-1-ontoβ†’(On βˆ– 𝐡))
21adantl 483 . . . . . 6 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (π‘₯ ∈ On ↦ (𝐡 +o π‘₯)):On–1-1-ontoβ†’(On βˆ– 𝐡))
3 f1of1 6787 . . . . . 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 7723 . . . . . 6 (𝐴 ∈ On β†’ 𝐴 βŠ† On)
65adantr 482 . . . . 5 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ 𝐴 βŠ† On)
7 f1ssres 6750 . . . . 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 585 . . . 4 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ ((π‘₯ ∈ On ↦ (𝐡 +o π‘₯)) β†Ύ 𝐴):𝐴–1-1β†’(On βˆ– 𝐡))
96resmptd 5998 . . . . . 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 2791 . . . . 5 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ ((π‘₯ ∈ On ↦ (𝐡 +o π‘₯)) β†Ύ 𝐴) = 𝐹)
12 f1eq1 6737 . . . . 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 6799 . . 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 6742 . . . 4 (𝐹:𝐴–1-1β†’(On βˆ– 𝐡) β†’ 𝐹:𝐴⟢(On βˆ– 𝐡))
18 frn 6679 . . . 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 4098 . . . 4 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ ran 𝐹 βŠ† On)
21 reldisj 4415 . . . 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 257 . 2 ((𝐴 ∈ On ∧ 𝐡 ∈ On) β†’ (ran 𝐹 ∩ 𝐡) = βˆ…)
2416, 23jca 513 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 397   = wceq 1542   ∈ wcel 2107   βˆ– cdif 3911   ∩ cin 3913   βŠ† wss 3914  βˆ…c0 4286   ↦ cmpt 5192  ran crn 5638   β†Ύ cres 5639  Oncon0 6321  βŸΆwf 6496  β€“1-1β†’wf1 6497  β€“1-1-ontoβ†’wf1o 6499  (class class class)co 7361   +o coa 8413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-oadd 8420
This theorem is referenced by:  oacomf1o  8516  onadju  10137
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