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Theorem ofpreima2 30578
Description: Express the preimage of a function operation as a union of preimages. This version of ofpreima 30577 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 30577 . . 3 (𝜑 → ((𝐹f 𝑅𝐺) “ 𝐷) = 𝑝 ∈ (𝑅𝐷)((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
6 inundif 4368 . . . . 5 (((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺)) ∪ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) = (𝑅𝐷)
7 iuneq1 4897 . . . . 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 4979 . . . 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 2763 . . 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, 10eqtrdi 2789 . 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 488 . . . . . . . . . . 11 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺)))
1312eldifbd 3856 . . . . . . . . . 10 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → ¬ 𝑝 ∈ (ran 𝐹 × ran 𝐺))
14 cnvimass 5923 . . . . . . . . . . . . . 14 (𝑅𝐷) ⊆ dom 𝑅
154fndmd 6442 . . . . . . . . . . . . . 14 (𝜑 → dom 𝑅 = (𝐵 × 𝐶))
1614, 15sseqtrid 3929 . . . . . . . . . . . . 13 (𝜑 → (𝑅𝐷) ⊆ (𝐵 × 𝐶))
1716ssdifssd 4033 . . . . . . . . . . . 12 (𝜑 → ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺)) ⊆ (𝐵 × 𝐶))
1817sselda 3877 . . . . . . . . . . 11 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (𝐵 × 𝐶))
19 1st2nd2 7753 . . . . . . . . . . 11 (𝑝 ∈ (𝐵 × 𝐶) → 𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩)
20 elxp6 7748 . . . . . . . . . . . 12 (𝑝 ∈ (ran 𝐹 × ran 𝐺) ↔ (𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩ ∧ ((1st𝑝) ∈ ran 𝐹 ∧ (2nd𝑝) ∈ ran 𝐺)))
2120simplbi2 504 . . . . . . . . . . 11 (𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩ → (((1st𝑝) ∈ ran 𝐹 ∧ (2nd𝑝) ∈ ran 𝐺) → 𝑝 ∈ (ran 𝐹 × ran 𝐺)))
2218, 19, 213syl 18 . . . . . . . . . 10 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → (((1st𝑝) ∈ ran 𝐹 ∧ (2nd𝑝) ∈ ran 𝐺) → 𝑝 ∈ (ran 𝐹 × ran 𝐺)))
2313, 22mtod 201 . . . . . . . . 9 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → ¬ ((1st𝑝) ∈ ran 𝐹 ∧ (2nd𝑝) ∈ ran 𝐺))
24 ianor 981 . . . . . . . . 9 (¬ ((1st𝑝) ∈ ran 𝐹 ∧ (2nd𝑝) ∈ ran 𝐺) ↔ (¬ (1st𝑝) ∈ ran 𝐹 ∨ ¬ (2nd𝑝) ∈ ran 𝐺))
2523, 24sylib 221 . . . . . . . 8 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → (¬ (1st𝑝) ∈ ran 𝐹 ∨ ¬ (2nd𝑝) ∈ ran 𝐺))
26 disjsn 4602 . . . . . . . . 9 ((ran 𝐹 ∩ {(1st𝑝)}) = ∅ ↔ ¬ (1st𝑝) ∈ ran 𝐹)
27 disjsn 4602 . . . . . . . . 9 ((ran 𝐺 ∩ {(2nd𝑝)}) = ∅ ↔ ¬ (2nd𝑝) ∈ ran 𝐺)
2826, 27orbi12i 914 . . . . . . . 8 (((ran 𝐹 ∩ {(1st𝑝)}) = ∅ ∨ (ran 𝐺 ∩ {(2nd𝑝)}) = ∅) ↔ (¬ (1st𝑝) ∈ ran 𝐹 ∨ ¬ (2nd𝑝) ∈ ran 𝐺))
2925, 28sylibr 237 . . . . . . 7 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → ((ran 𝐹 ∩ {(1st𝑝)}) = ∅ ∨ (ran 𝐺 ∩ {(2nd𝑝)}) = ∅))
301ffnd 6505 . . . . . . . . 9 (𝜑𝐹 Fn 𝐴)
31 dffn3 6517 . . . . . . . . 9 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
3230, 31sylib 221 . . . . . . . 8 (𝜑𝐹:𝐴⟶ran 𝐹)
332ffnd 6505 . . . . . . . . . 10 (𝜑𝐺 Fn 𝐴)
34 dffn3 6517 . . . . . . . . . 10 (𝐺 Fn 𝐴𝐺:𝐴⟶ran 𝐺)
3533, 34sylib 221 . . . . . . . . 9 (𝜑𝐺:𝐴⟶ran 𝐺)
3635adantr 484 . . . . . . . 8 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → 𝐺:𝐴⟶ran 𝐺)
37 fimacnvdisj 6556 . . . . . . . . . . 11 ((𝐹:𝐴⟶ran 𝐹 ∧ (ran 𝐹 ∩ {(1st𝑝)}) = ∅) → (𝐹 “ {(1st𝑝)}) = ∅)
38 ineq1 4096 . . . . . . . . . . . 12 ((𝐹 “ {(1st𝑝)}) = ∅ → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = (∅ ∩ (𝐺 “ {(2nd𝑝)})))
39 0in 4282 . . . . . . . . . . . 12 (∅ ∩ (𝐺 “ {(2nd𝑝)})) = ∅
4038, 39eqtrdi 2789 . . . . . . . . . . 11 ((𝐹 “ {(1st𝑝)}) = ∅ → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅)
4137, 40syl 17 . . . . . . . . . 10 ((𝐹:𝐴⟶ran 𝐹 ∧ (ran 𝐹 ∩ {(1st𝑝)}) = ∅) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅)
4241ex 416 . . . . . . . . 9 (𝐹:𝐴⟶ran 𝐹 → ((ran 𝐹 ∩ {(1st𝑝)}) = ∅ → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅))
43 fimacnvdisj 6556 . . . . . . . . . . 11 ((𝐺:𝐴⟶ran 𝐺 ∧ (ran 𝐺 ∩ {(2nd𝑝)}) = ∅) → (𝐺 “ {(2nd𝑝)}) = ∅)
44 ineq2 4097 . . . . . . . . . . . 12 ((𝐺 “ {(2nd𝑝)}) = ∅ → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ((𝐹 “ {(1st𝑝)}) ∩ ∅))
45 in0 4280 . . . . . . . . . . . 12 ((𝐹 “ {(1st𝑝)}) ∩ ∅) = ∅
4644, 45eqtrdi 2789 . . . . . . . . . . 11 ((𝐺 “ {(2nd𝑝)}) = ∅ → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅)
4743, 46syl 17 . . . . . . . . . 10 ((𝐺:𝐴⟶ran 𝐺 ∧ (ran 𝐺 ∩ {(2nd𝑝)}) = ∅) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅)
4847ex 416 . . . . . . . . 9 (𝐺:𝐴⟶ran 𝐺 → ((ran 𝐺 ∩ {(2nd𝑝)}) = ∅ → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅))
4942, 48jaao 954 . . . . . . . 8 ((𝐹:𝐴⟶ran 𝐹𝐺:𝐴⟶ran 𝐺) → (((ran 𝐹 ∩ {(1st𝑝)}) = ∅ ∨ (ran 𝐺 ∩ {(2nd𝑝)}) = ∅) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅))
5032, 36, 49syl2an2r 685 . . . . . . 7 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → (((ran 𝐹 ∩ {(1st𝑝)}) = ∅ ∨ (ran 𝐺 ∩ {(2nd𝑝)}) = ∅) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅))
5129, 50mpd 15 . . . . . 6 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅)
5251iuneq2dv 4905 . . . . 5 (𝜑 𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = 𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))∅)
53 iun0 4947 . . . . 5 𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))∅ = ∅
5452, 53eqtrdi 2789 . . . 4 (𝜑 𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅)
5554uneq2d 4053 . . 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)})) ∪ ∅))
56 un0 4279 . . 3 ( 𝑝 ∈ ((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∪ ∅) = 𝑝 ∈ ((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))
5755, 56eqtrdi 2789 . 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)})))
5811, 57eqtrd 2773 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  wa 399  wo 846   = wceq 1542  wcel 2114  cdif 3840  cun 3841  cin 3842  c0 4211  {csn 4516  cop 4522   ciun 4881   × cxp 5523  ccnv 5524  dom cdm 5525  ran crn 5526  cima 5528   Fn wfn 6334  wf 6335  cfv 6339  (class class class)co 7170  f cof 7423  1st c1st 7712  2nd c2nd 7713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-of 7425  df-1st 7714  df-2nd 7715
This theorem is referenced by:  sibfof  31877
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