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Theorem ofpreima2 29664
Description: Express the preimage of a function operation as a union of preimages. This version of ofpreima 29663 iterates the union over a smaller set. (Contributed by Thierry Arnoux, 8-Mar-2018.)
Hypotheses
Ref Expression
ofpreima.1 (𝜑𝐹:𝐴𝐵)
ofpreima.2 (𝜑𝐺:𝐴𝐶)
ofpreima.3 (𝜑𝐴𝑉)
ofpreima.4 (𝜑𝑅 Fn (𝐵 × 𝐶))
Assertion
Ref Expression
ofpreima2 (𝜑 → ((𝐹𝑓 𝑅𝐺) “ 𝐷) = 𝑝 ∈ ((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
Distinct variable groups:   𝐴,𝑝   𝐷,𝑝   𝐹,𝑝   𝐺,𝑝   𝑅,𝑝   𝜑,𝑝
Allowed substitution hints:   𝐵(𝑝)   𝐶(𝑝)   𝑉(𝑝)

Proof of Theorem ofpreima2
StepHypRef Expression
1 ofpreima.1 . . . 4 (𝜑𝐹:𝐴𝐵)
2 ofpreima.2 . . . 4 (𝜑𝐺:𝐴𝐶)
3 ofpreima.3 . . . 4 (𝜑𝐴𝑉)
4 ofpreima.4 . . . 4 (𝜑𝑅 Fn (𝐵 × 𝐶))
51, 2, 3, 4ofpreima 29663 . . 3 (𝜑 → ((𝐹𝑓 𝑅𝐺) “ 𝐷) = 𝑝 ∈ (𝑅𝐷)((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
6 inundif 4122 . . . . 5 (((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺)) ∪ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) = (𝑅𝐷)
7 iuneq1 4610 . . . . 5 ((((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺)) ∪ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) = (𝑅𝐷) → 𝑝 ∈ (((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺)) ∪ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺)))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = 𝑝 ∈ (𝑅𝐷)((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
86, 7ax-mp 5 . . . 4 𝑝 ∈ (((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺)) ∪ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺)))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = 𝑝 ∈ (𝑅𝐷)((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))
9 iunxun 4681 . . . 4 𝑝 ∈ (((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺)) ∪ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺)))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ( 𝑝 ∈ ((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∪ 𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
108, 9eqtr3i 2716 . . 3 𝑝 ∈ (𝑅𝐷)((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ( 𝑝 ∈ ((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∪ 𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
115, 10syl6eq 2742 . 2 (𝜑 → ((𝐹𝑓 𝑅𝐺) “ 𝐷) = ( 𝑝 ∈ ((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∪ 𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
12 simpr 479 . . . . . . . . . . 11 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺)))
1312eldifbd 3661 . . . . . . . . . 10 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → ¬ 𝑝 ∈ (ran 𝐹 × ran 𝐺))
14 cnvimass 5563 . . . . . . . . . . . . . 14 (𝑅𝐷) ⊆ dom 𝑅
15 fndm 6071 . . . . . . . . . . . . . . 15 (𝑅 Fn (𝐵 × 𝐶) → dom 𝑅 = (𝐵 × 𝐶))
164, 15syl 17 . . . . . . . . . . . . . 14 (𝜑 → dom 𝑅 = (𝐵 × 𝐶))
1714, 16syl5sseq 3727 . . . . . . . . . . . . 13 (𝜑 → (𝑅𝐷) ⊆ (𝐵 × 𝐶))
1817ssdifssd 3824 . . . . . . . . . . . 12 (𝜑 → ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺)) ⊆ (𝐵 × 𝐶))
1918sselda 3677 . . . . . . . . . . 11 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (𝐵 × 𝐶))
20 1st2nd2 7292 . . . . . . . . . . 11 (𝑝 ∈ (𝐵 × 𝐶) → 𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩)
21 elxp6 7287 . . . . . . . . . . . 12 (𝑝 ∈ (ran 𝐹 × ran 𝐺) ↔ (𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩ ∧ ((1st𝑝) ∈ ran 𝐹 ∧ (2nd𝑝) ∈ ran 𝐺)))
2221simplbi2 656 . . . . . . . . . . 11 (𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩ → (((1st𝑝) ∈ ran 𝐹 ∧ (2nd𝑝) ∈ ran 𝐺) → 𝑝 ∈ (ran 𝐹 × ran 𝐺)))
2319, 20, 223syl 18 . . . . . . . . . 10 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → (((1st𝑝) ∈ ran 𝐹 ∧ (2nd𝑝) ∈ ran 𝐺) → 𝑝 ∈ (ran 𝐹 × ran 𝐺)))
2413, 23mtod 189 . . . . . . . . 9 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → ¬ ((1st𝑝) ∈ ran 𝐹 ∧ (2nd𝑝) ∈ ran 𝐺))
25 ianor 510 . . . . . . . . 9 (¬ ((1st𝑝) ∈ ran 𝐹 ∧ (2nd𝑝) ∈ ran 𝐺) ↔ (¬ (1st𝑝) ∈ ran 𝐹 ∨ ¬ (2nd𝑝) ∈ ran 𝐺))
2624, 25sylib 208 . . . . . . . 8 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → (¬ (1st𝑝) ∈ ran 𝐹 ∨ ¬ (2nd𝑝) ∈ ran 𝐺))
27 disjsn 4321 . . . . . . . . 9 ((ran 𝐹 ∩ {(1st𝑝)}) = ∅ ↔ ¬ (1st𝑝) ∈ ran 𝐹)
28 disjsn 4321 . . . . . . . . 9 ((ran 𝐺 ∩ {(2nd𝑝)}) = ∅ ↔ ¬ (2nd𝑝) ∈ ran 𝐺)
2927, 28orbi12i 544 . . . . . . . 8 (((ran 𝐹 ∩ {(1st𝑝)}) = ∅ ∨ (ran 𝐺 ∩ {(2nd𝑝)}) = ∅) ↔ (¬ (1st𝑝) ∈ ran 𝐹 ∨ ¬ (2nd𝑝) ∈ ran 𝐺))
3026, 29sylibr 224 . . . . . . 7 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → ((ran 𝐹 ∩ {(1st𝑝)}) = ∅ ∨ (ran 𝐺 ∩ {(2nd𝑝)}) = ∅))
311ffnd 6127 . . . . . . . . 9 (𝜑𝐹 Fn 𝐴)
32 dffn3 6135 . . . . . . . . 9 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
3331, 32sylib 208 . . . . . . . 8 (𝜑𝐹:𝐴⟶ran 𝐹)
342ffnd 6127 . . . . . . . . . 10 (𝜑𝐺 Fn 𝐴)
35 dffn3 6135 . . . . . . . . . 10 (𝐺 Fn 𝐴𝐺:𝐴⟶ran 𝐺)
3634, 35sylib 208 . . . . . . . . 9 (𝜑𝐺:𝐴⟶ran 𝐺)
3736adantr 472 . . . . . . . 8 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → 𝐺:𝐴⟶ran 𝐺)
38 fimacnvdisj 6164 . . . . . . . . . . 11 ((𝐹:𝐴⟶ran 𝐹 ∧ (ran 𝐹 ∩ {(1st𝑝)}) = ∅) → (𝐹 “ {(1st𝑝)}) = ∅)
39 ineq1 3883 . . . . . . . . . . . 12 ((𝐹 “ {(1st𝑝)}) = ∅ → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = (∅ ∩ (𝐺 “ {(2nd𝑝)})))
40 0in 4045 . . . . . . . . . . . 12 (∅ ∩ (𝐺 “ {(2nd𝑝)})) = ∅
4139, 40syl6eq 2742 . . . . . . . . . . 11 ((𝐹 “ {(1st𝑝)}) = ∅ → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅)
4238, 41syl 17 . . . . . . . . . 10 ((𝐹:𝐴⟶ran 𝐹 ∧ (ran 𝐹 ∩ {(1st𝑝)}) = ∅) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅)
4342ex 449 . . . . . . . . 9 (𝐹:𝐴⟶ran 𝐹 → ((ran 𝐹 ∩ {(1st𝑝)}) = ∅ → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅))
44 fimacnvdisj 6164 . . . . . . . . . . 11 ((𝐺:𝐴⟶ran 𝐺 ∧ (ran 𝐺 ∩ {(2nd𝑝)}) = ∅) → (𝐺 “ {(2nd𝑝)}) = ∅)
45 ineq2 3884 . . . . . . . . . . . 12 ((𝐺 “ {(2nd𝑝)}) = ∅ → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ((𝐹 “ {(1st𝑝)}) ∩ ∅))
46 in0 4044 . . . . . . . . . . . 12 ((𝐹 “ {(1st𝑝)}) ∩ ∅) = ∅
4745, 46syl6eq 2742 . . . . . . . . . . 11 ((𝐺 “ {(2nd𝑝)}) = ∅ → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅)
4844, 47syl 17 . . . . . . . . . 10 ((𝐺:𝐴⟶ran 𝐺 ∧ (ran 𝐺 ∩ {(2nd𝑝)}) = ∅) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅)
4948ex 449 . . . . . . . . 9 (𝐺:𝐴⟶ran 𝐺 → ((ran 𝐺 ∩ {(2nd𝑝)}) = ∅ → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅))
5043, 49jaao 532 . . . . . . . 8 ((𝐹:𝐴⟶ran 𝐹𝐺:𝐴⟶ran 𝐺) → (((ran 𝐹 ∩ {(1st𝑝)}) = ∅ ∨ (ran 𝐺 ∩ {(2nd𝑝)}) = ∅) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅))
5133, 37, 50syl2an2r 911 . . . . . . 7 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → (((ran 𝐹 ∩ {(1st𝑝)}) = ∅ ∨ (ran 𝐺 ∩ {(2nd𝑝)}) = ∅) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅))
5230, 51mpd 15 . . . . . 6 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅)
5352iuneq2dv 4618 . . . . 5 (𝜑 𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = 𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))∅)
54 iun0 4652 . . . . 5 𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))∅ = ∅
5553, 54syl6eq 2742 . . . 4 (𝜑 𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅)
5655uneq2d 3843 . . 3 (𝜑 → ( 𝑝 ∈ ((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∪ 𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))) = ( 𝑝 ∈ ((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∪ ∅))
57 un0 4043 . . 3 ( 𝑝 ∈ ((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∪ ∅) = 𝑝 ∈ ((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))
5856, 57syl6eq 2742 . 2 (𝜑 → ( 𝑝 ∈ ((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∪ 𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))) = 𝑝 ∈ ((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
5911, 58eqtrd 2726 1 (𝜑 → ((𝐹𝑓 𝑅𝐺) “ 𝐷) = 𝑝 ∈ ((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  wa 383   = wceq 1564  wcel 2071  cdif 3645  cun 3646  cin 3647  c0 3991  {csn 4253  cop 4259   ciun 4596   × cxp 5184  ccnv 5185  dom cdm 5186  ran crn 5187  cima 5189   Fn wfn 5964  wf 5965  cfv 5969  (class class class)co 6733  𝑓 cof 6980  1st c1st 7251  2nd c2nd 7252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1818  ax-5 1920  ax-6 1986  ax-7 2022  ax-8 2073  ax-9 2080  ax-10 2100  ax-11 2115  ax-12 2128  ax-13 2323  ax-ext 2672  ax-rep 4847  ax-sep 4857  ax-nul 4865  ax-pow 4916  ax-pr 4979  ax-un 7034
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1567  df-ex 1786  df-nf 1791  df-sb 1979  df-eu 2543  df-mo 2544  df-clab 2679  df-cleq 2685  df-clel 2688  df-nfc 2823  df-ne 2865  df-ral 2987  df-rex 2988  df-reu 2989  df-rab 2991  df-v 3274  df-sbc 3510  df-csb 3608  df-dif 3651  df-un 3653  df-in 3655  df-ss 3662  df-nul 3992  df-if 4163  df-sn 4254  df-pr 4256  df-op 4260  df-uni 4513  df-iun 4598  df-br 4729  df-opab 4789  df-mpt 4806  df-id 5096  df-xp 5192  df-rel 5193  df-cnv 5194  df-co 5195  df-dm 5196  df-rn 5197  df-res 5198  df-ima 5199  df-iota 5932  df-fun 5971  df-fn 5972  df-f 5973  df-f1 5974  df-fo 5975  df-f1o 5976  df-fv 5977  df-ov 6736  df-oprab 6737  df-mpt2 6738  df-of 6982  df-1st 7253  df-2nd 7254
This theorem is referenced by:  sibfof  30600
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