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Theorem fnrelpredd 32535
 Description: A function that preserves a relation also preserves predecessors. (Contributed by BTernaryTau, 16-Jul-2024.)
Hypotheses
Ref Expression
fnrelpredd.1 (𝜑𝐹 Fn 𝐴)
fnrelpredd.2 (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐹𝑥)𝑆(𝐹𝑦)))
fnrelpredd.3 (𝜑𝐶𝐴)
fnrelpredd.4 (𝜑𝐷𝐴)
Assertion
Ref Expression
fnrelpredd (𝜑 → Pred(𝑆, (𝐹𝐶), (𝐹𝐷)) = (𝐹 “ Pred(𝑅, 𝐶, 𝐷)))
Distinct variable groups:   𝑦,𝐴   𝜑,𝑥   𝑥,𝐷,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦   𝑥,𝐹,𝑦   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑥)   𝐶(𝑦)

Proof of Theorem fnrelpredd
Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6665 . . . . 5 (𝐹𝐷) ∈ V
21dfpred3 6131 . . . 4 Pred(𝑆, (𝐹𝐶), (𝐹𝐷)) = {𝑣 ∈ (𝐹𝐶) ∣ 𝑣𝑆(𝐹𝐷)}
3 elrabi 3623 . . . . . . . . . . 11 (𝑢 ∈ {𝑥𝐶 ∣ (𝐹𝑥)𝑆(𝐹𝐷)} → 𝑢𝐶)
43anim1i 617 . . . . . . . . . 10 ((𝑢 ∈ {𝑥𝐶 ∣ (𝐹𝑥)𝑆(𝐹𝐷)} ∧ (𝐹𝑢) = 𝑣) → (𝑢𝐶 ∧ (𝐹𝑢) = 𝑣))
54reximi2 3207 . . . . . . . . 9 (∃𝑢 ∈ {𝑥𝐶 ∣ (𝐹𝑥)𝑆(𝐹𝐷)} (𝐹𝑢) = 𝑣 → ∃𝑢𝐶 (𝐹𝑢) = 𝑣)
6 fnrelpredd.1 . . . . . . . . . 10 (𝜑𝐹 Fn 𝐴)
7 fnrelpredd.3 . . . . . . . . . 10 (𝜑𝐶𝐴)
86, 7fvelimabd 6720 . . . . . . . . 9 (𝜑 → (𝑣 ∈ (𝐹𝐶) ↔ ∃𝑢𝐶 (𝐹𝑢) = 𝑣))
95, 8syl5ibr 249 . . . . . . . 8 (𝜑 → (∃𝑢 ∈ {𝑥𝐶 ∣ (𝐹𝑥)𝑆(𝐹𝐷)} (𝐹𝑢) = 𝑣𝑣 ∈ (𝐹𝐶)))
10 fveq2 6652 . . . . . . . . . . . 12 (𝑥 = 𝑢 → (𝐹𝑥) = (𝐹𝑢))
1110breq1d 5043 . . . . . . . . . . 11 (𝑥 = 𝑢 → ((𝐹𝑥)𝑆(𝐹𝐷) ↔ (𝐹𝑢)𝑆(𝐹𝐷)))
1211elrab 3629 . . . . . . . . . 10 (𝑢 ∈ {𝑥𝐶 ∣ (𝐹𝑥)𝑆(𝐹𝐷)} ↔ (𝑢𝐶 ∧ (𝐹𝑢)𝑆(𝐹𝐷)))
13 breq1 5036 . . . . . . . . . . . 12 ((𝐹𝑢) = 𝑣 → ((𝐹𝑢)𝑆(𝐹𝐷) ↔ 𝑣𝑆(𝐹𝐷)))
1413biimpac 482 . . . . . . . . . . 11 (((𝐹𝑢)𝑆(𝐹𝐷) ∧ (𝐹𝑢) = 𝑣) → 𝑣𝑆(𝐹𝐷))
1514adantll 713 . . . . . . . . . 10 (((𝑢𝐶 ∧ (𝐹𝑢)𝑆(𝐹𝐷)) ∧ (𝐹𝑢) = 𝑣) → 𝑣𝑆(𝐹𝐷))
1612, 15sylanb 584 . . . . . . . . 9 ((𝑢 ∈ {𝑥𝐶 ∣ (𝐹𝑥)𝑆(𝐹𝐷)} ∧ (𝐹𝑢) = 𝑣) → 𝑣𝑆(𝐹𝐷))
1716rexlimiva 3240 . . . . . . . 8 (∃𝑢 ∈ {𝑥𝐶 ∣ (𝐹𝑥)𝑆(𝐹𝐷)} (𝐹𝑢) = 𝑣𝑣𝑆(𝐹𝐷))
189, 17jca2 517 . . . . . . 7 (𝜑 → (∃𝑢 ∈ {𝑥𝐶 ∣ (𝐹𝑥)𝑆(𝐹𝐷)} (𝐹𝑢) = 𝑣 → (𝑣 ∈ (𝐹𝐶) ∧ 𝑣𝑆(𝐹𝐷))))
198biimpd 232 . . . . . . . . 9 (𝜑 → (𝑣 ∈ (𝐹𝐶) → ∃𝑢𝐶 (𝐹𝑢) = 𝑣))
2019adantrd 495 . . . . . . . 8 (𝜑 → ((𝑣 ∈ (𝐹𝐶) ∧ 𝑣𝑆(𝐹𝐷)) → ∃𝑢𝐶 (𝐹𝑢) = 𝑣))
21 simpl 486 . . . . . . . . . . . . 13 ((𝑢𝐶 ∧ (𝐹𝑢) = 𝑣) → 𝑢𝐶)
2221a1i 11 . . . . . . . . . . . 12 (𝑣𝑆(𝐹𝐷) → ((𝑢𝐶 ∧ (𝐹𝑢) = 𝑣) → 𝑢𝐶))
2313biimprcd 253 . . . . . . . . . . . . 13 (𝑣𝑆(𝐹𝐷) → ((𝐹𝑢) = 𝑣 → (𝐹𝑢)𝑆(𝐹𝐷)))
2423adantld 494 . . . . . . . . . . . 12 (𝑣𝑆(𝐹𝐷) → ((𝑢𝐶 ∧ (𝐹𝑢) = 𝑣) → (𝐹𝑢)𝑆(𝐹𝐷)))
25 simpr 488 . . . . . . . . . . . . 13 ((𝑢𝐶 ∧ (𝐹𝑢) = 𝑣) → (𝐹𝑢) = 𝑣)
2625a1i 11 . . . . . . . . . . . 12 (𝑣𝑆(𝐹𝐷) → ((𝑢𝐶 ∧ (𝐹𝑢) = 𝑣) → (𝐹𝑢) = 𝑣))
2722, 24, 263jcad 1126 . . . . . . . . . . 11 (𝑣𝑆(𝐹𝐷) → ((𝑢𝐶 ∧ (𝐹𝑢) = 𝑣) → (𝑢𝐶 ∧ (𝐹𝑢)𝑆(𝐹𝐷) ∧ (𝐹𝑢) = 𝑣)))
2812biimpri 231 . . . . . . . . . . . . 13 ((𝑢𝐶 ∧ (𝐹𝑢)𝑆(𝐹𝐷)) → 𝑢 ∈ {𝑥𝐶 ∣ (𝐹𝑥)𝑆(𝐹𝐷)})
2928anim1i 617 . . . . . . . . . . . 12 (((𝑢𝐶 ∧ (𝐹𝑢)𝑆(𝐹𝐷)) ∧ (𝐹𝑢) = 𝑣) → (𝑢 ∈ {𝑥𝐶 ∣ (𝐹𝑥)𝑆(𝐹𝐷)} ∧ (𝐹𝑢) = 𝑣))
30293impa 1107 . . . . . . . . . . 11 ((𝑢𝐶 ∧ (𝐹𝑢)𝑆(𝐹𝐷) ∧ (𝐹𝑢) = 𝑣) → (𝑢 ∈ {𝑥𝐶 ∣ (𝐹𝑥)𝑆(𝐹𝐷)} ∧ (𝐹𝑢) = 𝑣))
3127, 30syl6 35 . . . . . . . . . 10 (𝑣𝑆(𝐹𝐷) → ((𝑢𝐶 ∧ (𝐹𝑢) = 𝑣) → (𝑢 ∈ {𝑥𝐶 ∣ (𝐹𝑥)𝑆(𝐹𝐷)} ∧ (𝐹𝑢) = 𝑣)))
3231reximdv2 3230 . . . . . . . . 9 (𝑣𝑆(𝐹𝐷) → (∃𝑢𝐶 (𝐹𝑢) = 𝑣 → ∃𝑢 ∈ {𝑥𝐶 ∣ (𝐹𝑥)𝑆(𝐹𝐷)} (𝐹𝑢) = 𝑣))
3332adantl 485 . . . . . . . 8 ((𝑣 ∈ (𝐹𝐶) ∧ 𝑣𝑆(𝐹𝐷)) → (∃𝑢𝐶 (𝐹𝑢) = 𝑣 → ∃𝑢 ∈ {𝑥𝐶 ∣ (𝐹𝑥)𝑆(𝐹𝐷)} (𝐹𝑢) = 𝑣))
3420, 33sylcom 30 . . . . . . 7 (𝜑 → ((𝑣 ∈ (𝐹𝐶) ∧ 𝑣𝑆(𝐹𝐷)) → ∃𝑢 ∈ {𝑥𝐶 ∣ (𝐹𝑥)𝑆(𝐹𝐷)} (𝐹𝑢) = 𝑣))
3518, 34impbid 215 . . . . . 6 (𝜑 → (∃𝑢 ∈ {𝑥𝐶 ∣ (𝐹𝑥)𝑆(𝐹𝐷)} (𝐹𝑢) = 𝑣 ↔ (𝑣 ∈ (𝐹𝐶) ∧ 𝑣𝑆(𝐹𝐷))))
3635abbidv 2862 . . . . 5 (𝜑 → {𝑣 ∣ ∃𝑢 ∈ {𝑥𝐶 ∣ (𝐹𝑥)𝑆(𝐹𝐷)} (𝐹𝑢) = 𝑣} = {𝑣 ∣ (𝑣 ∈ (𝐹𝐶) ∧ 𝑣𝑆(𝐹𝐷))})
37 df-rab 3115 . . . . 5 {𝑣 ∈ (𝐹𝐶) ∣ 𝑣𝑆(𝐹𝐷)} = {𝑣 ∣ (𝑣 ∈ (𝐹𝐶) ∧ 𝑣𝑆(𝐹𝐷))}
3836, 37eqtr4di 2851 . . . 4 (𝜑 → {𝑣 ∣ ∃𝑢 ∈ {𝑥𝐶 ∣ (𝐹𝑥)𝑆(𝐹𝐷)} (𝐹𝑢) = 𝑣} = {𝑣 ∈ (𝐹𝐶) ∣ 𝑣𝑆(𝐹𝐷)})
392, 38eqtr4id 2852 . . 3 (𝜑 → Pred(𝑆, (𝐹𝐶), (𝐹𝐷)) = {𝑣 ∣ ∃𝑢 ∈ {𝑥𝐶 ∣ (𝐹𝑥)𝑆(𝐹𝐷)} (𝐹𝑢) = 𝑣})
40 fnfun 6428 . . . . 5 (𝐹 Fn 𝐴 → Fun 𝐹)
416, 40syl 17 . . . 4 (𝜑 → Fun 𝐹)
42 ssrab2 4008 . . . . . 6 {𝑥𝐶 ∣ (𝐹𝑥)𝑆(𝐹𝐷)} ⊆ 𝐶
4342, 7sstrid 3927 . . . . 5 (𝜑 → {𝑥𝐶 ∣ (𝐹𝑥)𝑆(𝐹𝐷)} ⊆ 𝐴)
446fndmd 6432 . . . . 5 (𝜑 → dom 𝐹 = 𝐴)
4543, 44sseqtrrd 3957 . . . 4 (𝜑 → {𝑥𝐶 ∣ (𝐹𝑥)𝑆(𝐹𝐷)} ⊆ dom 𝐹)
46 dfimafn 6710 . . . 4 ((Fun 𝐹 ∧ {𝑥𝐶 ∣ (𝐹𝑥)𝑆(𝐹𝐷)} ⊆ dom 𝐹) → (𝐹 “ {𝑥𝐶 ∣ (𝐹𝑥)𝑆(𝐹𝐷)}) = {𝑣 ∣ ∃𝑢 ∈ {𝑥𝐶 ∣ (𝐹𝑥)𝑆(𝐹𝐷)} (𝐹𝑢) = 𝑣})
4741, 45, 46syl2anc 587 . . 3 (𝜑 → (𝐹 “ {𝑥𝐶 ∣ (𝐹𝑥)𝑆(𝐹𝐷)}) = {𝑣 ∣ ∃𝑢 ∈ {𝑥𝐶 ∣ (𝐹𝑥)𝑆(𝐹𝐷)} (𝐹𝑢) = 𝑣})
4839, 47eqtr4d 2836 . 2 (𝜑 → Pred(𝑆, (𝐹𝐶), (𝐹𝐷)) = (𝐹 “ {𝑥𝐶 ∣ (𝐹𝑥)𝑆(𝐹𝐷)}))
49 fnrelpredd.4 . . . . 5 (𝜑𝐷𝐴)
50 dfpred3g 6132 . . . . 5 (𝐷𝐴 → Pred(𝑅, 𝐶, 𝐷) = {𝑥𝐶𝑥𝑅𝐷})
5149, 50syl 17 . . . 4 (𝜑 → Pred(𝑅, 𝐶, 𝐷) = {𝑥𝐶𝑥𝑅𝐷})
527sselda 3916 . . . . . 6 ((𝜑𝑥𝐶) → 𝑥𝐴)
53 fnrelpredd.2 . . . . . . . 8 (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐹𝑥)𝑆(𝐹𝑦)))
5453r19.21bi 3173 . . . . . . 7 ((𝜑𝑥𝐴) → ∀𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐹𝑥)𝑆(𝐹𝑦)))
55 breq2 5037 . . . . . . . . . . 11 (𝑦 = 𝐷 → (𝑥𝑅𝑦𝑥𝑅𝐷))
56 fveq2 6652 . . . . . . . . . . . 12 (𝑦 = 𝐷 → (𝐹𝑦) = (𝐹𝐷))
5756breq2d 5045 . . . . . . . . . . 11 (𝑦 = 𝐷 → ((𝐹𝑥)𝑆(𝐹𝑦) ↔ (𝐹𝑥)𝑆(𝐹𝐷)))
5855, 57bibi12d 349 . . . . . . . . . 10 (𝑦 = 𝐷 → ((𝑥𝑅𝑦 ↔ (𝐹𝑥)𝑆(𝐹𝑦)) ↔ (𝑥𝑅𝐷 ↔ (𝐹𝑥)𝑆(𝐹𝐷))))
5958rspcv 3566 . . . . . . . . 9 (𝐷𝐴 → (∀𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐹𝑥)𝑆(𝐹𝑦)) → (𝑥𝑅𝐷 ↔ (𝐹𝑥)𝑆(𝐹𝐷))))
6049, 59syl 17 . . . . . . . 8 (𝜑 → (∀𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐹𝑥)𝑆(𝐹𝑦)) → (𝑥𝑅𝐷 ↔ (𝐹𝑥)𝑆(𝐹𝐷))))
6160adantr 484 . . . . . . 7 ((𝜑𝑥𝐴) → (∀𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐹𝑥)𝑆(𝐹𝑦)) → (𝑥𝑅𝐷 ↔ (𝐹𝑥)𝑆(𝐹𝐷))))
6254, 61mpd 15 . . . . . 6 ((𝜑𝑥𝐴) → (𝑥𝑅𝐷 ↔ (𝐹𝑥)𝑆(𝐹𝐷)))
6352, 62syldan 594 . . . . 5 ((𝜑𝑥𝐶) → (𝑥𝑅𝐷 ↔ (𝐹𝑥)𝑆(𝐹𝐷)))
6463rabbidva 3425 . . . 4 (𝜑 → {𝑥𝐶𝑥𝑅𝐷} = {𝑥𝐶 ∣ (𝐹𝑥)𝑆(𝐹𝐷)})
6551, 64eqtrd 2833 . . 3 (𝜑 → Pred(𝑅, 𝐶, 𝐷) = {𝑥𝐶 ∣ (𝐹𝑥)𝑆(𝐹𝐷)})
6665imaeq2d 5899 . 2 (𝜑 → (𝐹 “ Pred(𝑅, 𝐶, 𝐷)) = (𝐹 “ {𝑥𝐶 ∣ (𝐹𝑥)𝑆(𝐹𝐷)}))
6748, 66eqtr4d 2836 1 (𝜑 → Pred(𝑆, (𝐹𝐶), (𝐹𝐷)) = (𝐹 “ Pred(𝑅, 𝐶, 𝐷)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  {cab 2776  ∀wral 3106  ∃wrex 3107  {crab 3110   ⊆ wss 3882   class class class wbr 5033  dom cdm 5522   “ cima 5525  Predcpred 6120  Fun wfun 6323   Fn wfn 6324  ‘cfv 6329 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3722  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-nul 4246  df-if 4428  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-pred 6121  df-iota 6288  df-fun 6331  df-fn 6332  df-fv 6337 This theorem is referenced by:  cardpred  32536
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