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Theorem suppimacnv 7457
Description: Support sets of functions expressed by inverse images. (Contributed by AV, 31-Mar-2019.) (Revised by AV, 7-Apr-2019.)
Assertion
Ref Expression
suppimacnv ((𝑅𝑉𝑍𝑊) → (𝑅 supp 𝑍) = (𝑅 “ (V ∖ {𝑍})))

Proof of Theorem suppimacnv
Dummy variables 𝑥 𝑦 𝑠 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq2 4790 . . . . . . . 8 (𝑡 = 𝑠 → (𝑥𝑅𝑡𝑥𝑅𝑠))
21cbvexvw 2126 . . . . . . 7 (∃𝑡 𝑥𝑅𝑡 ↔ ∃𝑠 𝑥𝑅𝑠)
3 breq2 4790 . . . . . . . . . . . . . 14 (𝑠 = 𝑍 → (𝑥𝑅𝑠𝑥𝑅𝑍))
43anbi1d 615 . . . . . . . . . . . . 13 (𝑠 = 𝑍 → ((𝑥𝑅𝑠 ∧ (𝑥𝑅𝑡𝑡𝑍)) ↔ (𝑥𝑅𝑍 ∧ (𝑥𝑅𝑡𝑡𝑍))))
5 bianir 1045 . . . . . . . . . . . . . . . . . 18 ((𝑡𝑍 ∧ (𝑥𝑅𝑡𝑡𝑍)) → 𝑥𝑅𝑡)
6 vex 3354 . . . . . . . . . . . . . . . . . . . 20 𝑡 ∈ V
7 breq2 4790 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑡 → (𝑥𝑅𝑦𝑥𝑅𝑡))
8 neeq1 3005 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑡 → (𝑦𝑍𝑡𝑍))
97, 8anbi12d 616 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑡 → ((𝑥𝑅𝑦𝑦𝑍) ↔ (𝑥𝑅𝑡𝑡𝑍)))
106, 9spcev 3451 . . . . . . . . . . . . . . . . . . 19 ((𝑥𝑅𝑡𝑡𝑍) → ∃𝑦(𝑥𝑅𝑦𝑦𝑍))
1110ex 397 . . . . . . . . . . . . . . . . . 18 (𝑥𝑅𝑡 → (𝑡𝑍 → ∃𝑦(𝑥𝑅𝑦𝑦𝑍)))
125, 11syl 17 . . . . . . . . . . . . . . . . 17 ((𝑡𝑍 ∧ (𝑥𝑅𝑡𝑡𝑍)) → (𝑡𝑍 → ∃𝑦(𝑥𝑅𝑦𝑦𝑍)))
1312ex 397 . . . . . . . . . . . . . . . 16 (𝑡𝑍 → ((𝑥𝑅𝑡𝑡𝑍) → (𝑡𝑍 → ∃𝑦(𝑥𝑅𝑦𝑦𝑍))))
1413pm2.43a 54 . . . . . . . . . . . . . . 15 (𝑡𝑍 → ((𝑥𝑅𝑡𝑡𝑍) → ∃𝑦(𝑥𝑅𝑦𝑦𝑍)))
1514adantld 478 . . . . . . . . . . . . . 14 (𝑡𝑍 → ((𝑥𝑅𝑍 ∧ (𝑥𝑅𝑡𝑡𝑍)) → ∃𝑦(𝑥𝑅𝑦𝑦𝑍)))
16 nne 2947 . . . . . . . . . . . . . . . 16 𝑡𝑍𝑡 = 𝑍)
17 notbi 308 . . . . . . . . . . . . . . . . . . . 20 ((𝑥𝑅𝑡𝑡𝑍) ↔ (¬ 𝑥𝑅𝑡 ↔ ¬ 𝑡𝑍))
18 bianir 1045 . . . . . . . . . . . . . . . . . . . . . 22 ((¬ 𝑡𝑍 ∧ (¬ 𝑥𝑅𝑡 ↔ ¬ 𝑡𝑍)) → ¬ 𝑥𝑅𝑡)
19 breq2 4790 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑍 = 𝑡 → (𝑥𝑅𝑍𝑥𝑅𝑡))
2019eqcoms 2779 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑍 → (𝑥𝑅𝑍𝑥𝑅𝑡))
21 pm2.24 122 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥𝑅𝑡 → (¬ 𝑥𝑅𝑡 → ∃𝑦(𝑥𝑅𝑦𝑦𝑍)))
2220, 21syl6bi 243 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑍 → (𝑥𝑅𝑍 → (¬ 𝑥𝑅𝑡 → ∃𝑦(𝑥𝑅𝑦𝑦𝑍))))
2322com13 88 . . . . . . . . . . . . . . . . . . . . . 22 𝑥𝑅𝑡 → (𝑥𝑅𝑍 → (𝑡 = 𝑍 → ∃𝑦(𝑥𝑅𝑦𝑦𝑍))))
2418, 23syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((¬ 𝑡𝑍 ∧ (¬ 𝑥𝑅𝑡 ↔ ¬ 𝑡𝑍)) → (𝑥𝑅𝑍 → (𝑡 = 𝑍 → ∃𝑦(𝑥𝑅𝑦𝑦𝑍))))
2524ex 397 . . . . . . . . . . . . . . . . . . . 20 𝑡𝑍 → ((¬ 𝑥𝑅𝑡 ↔ ¬ 𝑡𝑍) → (𝑥𝑅𝑍 → (𝑡 = 𝑍 → ∃𝑦(𝑥𝑅𝑦𝑦𝑍)))))
2617, 25syl5bi 232 . . . . . . . . . . . . . . . . . . 19 𝑡𝑍 → ((𝑥𝑅𝑡𝑡𝑍) → (𝑥𝑅𝑍 → (𝑡 = 𝑍 → ∃𝑦(𝑥𝑅𝑦𝑦𝑍)))))
2726com13 88 . . . . . . . . . . . . . . . . . 18 (𝑥𝑅𝑍 → ((𝑥𝑅𝑡𝑡𝑍) → (¬ 𝑡𝑍 → (𝑡 = 𝑍 → ∃𝑦(𝑥𝑅𝑦𝑦𝑍)))))
2827imp 393 . . . . . . . . . . . . . . . . 17 ((𝑥𝑅𝑍 ∧ (𝑥𝑅𝑡𝑡𝑍)) → (¬ 𝑡𝑍 → (𝑡 = 𝑍 → ∃𝑦(𝑥𝑅𝑦𝑦𝑍))))
2928com13 88 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑍 → (¬ 𝑡𝑍 → ((𝑥𝑅𝑍 ∧ (𝑥𝑅𝑡𝑡𝑍)) → ∃𝑦(𝑥𝑅𝑦𝑦𝑍))))
3016, 29sylbi 207 . . . . . . . . . . . . . . 15 𝑡𝑍 → (¬ 𝑡𝑍 → ((𝑥𝑅𝑍 ∧ (𝑥𝑅𝑡𝑡𝑍)) → ∃𝑦(𝑥𝑅𝑦𝑦𝑍))))
3130pm2.43i 52 . . . . . . . . . . . . . 14 𝑡𝑍 → ((𝑥𝑅𝑍 ∧ (𝑥𝑅𝑡𝑡𝑍)) → ∃𝑦(𝑥𝑅𝑦𝑦𝑍)))
3215, 31pm2.61i 176 . . . . . . . . . . . . 13 ((𝑥𝑅𝑍 ∧ (𝑥𝑅𝑡𝑡𝑍)) → ∃𝑦(𝑥𝑅𝑦𝑦𝑍))
334, 32syl6bi 243 . . . . . . . . . . . 12 (𝑠 = 𝑍 → ((𝑥𝑅𝑠 ∧ (𝑥𝑅𝑡𝑡𝑍)) → ∃𝑦(𝑥𝑅𝑦𝑦𝑍)))
34 vex 3354 . . . . . . . . . . . . . . . 16 𝑠 ∈ V
35 breq2 4790 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑠 → (𝑥𝑅𝑦𝑥𝑅𝑠))
36 neeq1 3005 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑠 → (𝑦𝑍𝑠𝑍))
3735, 36anbi12d 616 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑠 → ((𝑥𝑅𝑦𝑦𝑍) ↔ (𝑥𝑅𝑠𝑠𝑍)))
3834, 37spcev 3451 . . . . . . . . . . . . . . 15 ((𝑥𝑅𝑠𝑠𝑍) → ∃𝑦(𝑥𝑅𝑦𝑦𝑍))
3938ex 397 . . . . . . . . . . . . . 14 (𝑥𝑅𝑠 → (𝑠𝑍 → ∃𝑦(𝑥𝑅𝑦𝑦𝑍)))
4039adantr 466 . . . . . . . . . . . . 13 ((𝑥𝑅𝑠 ∧ (𝑥𝑅𝑡𝑡𝑍)) → (𝑠𝑍 → ∃𝑦(𝑥𝑅𝑦𝑦𝑍)))
4140com12 32 . . . . . . . . . . . 12 (𝑠𝑍 → ((𝑥𝑅𝑠 ∧ (𝑥𝑅𝑡𝑡𝑍)) → ∃𝑦(𝑥𝑅𝑦𝑦𝑍)))
4233, 41pm2.61ine 3026 . . . . . . . . . . 11 ((𝑥𝑅𝑠 ∧ (𝑥𝑅𝑡𝑡𝑍)) → ∃𝑦(𝑥𝑅𝑦𝑦𝑍))
4342expcom 398 . . . . . . . . . 10 ((𝑥𝑅𝑡𝑡𝑍) → (𝑥𝑅𝑠 → ∃𝑦(𝑥𝑅𝑦𝑦𝑍)))
4443exlimiv 2010 . . . . . . . . 9 (∃𝑡(𝑥𝑅𝑡𝑡𝑍) → (𝑥𝑅𝑠 → ∃𝑦(𝑥𝑅𝑦𝑦𝑍)))
4544com12 32 . . . . . . . 8 (𝑥𝑅𝑠 → (∃𝑡(𝑥𝑅𝑡𝑡𝑍) → ∃𝑦(𝑥𝑅𝑦𝑦𝑍)))
4645exlimiv 2010 . . . . . . 7 (∃𝑠 𝑥𝑅𝑠 → (∃𝑡(𝑥𝑅𝑡𝑡𝑍) → ∃𝑦(𝑥𝑅𝑦𝑦𝑍)))
472, 46sylbi 207 . . . . . 6 (∃𝑡 𝑥𝑅𝑡 → (∃𝑡(𝑥𝑅𝑡𝑡𝑍) → ∃𝑦(𝑥𝑅𝑦𝑦𝑍)))
4847imp 393 . . . . 5 ((∃𝑡 𝑥𝑅𝑡 ∧ ∃𝑡(𝑥𝑅𝑡𝑡𝑍)) → ∃𝑦(𝑥𝑅𝑦𝑦𝑍))
4948a1i 11 . . . 4 ((𝑅𝑉𝑍𝑊) → ((∃𝑡 𝑥𝑅𝑡 ∧ ∃𝑡(𝑥𝑅𝑡𝑡𝑍)) → ∃𝑦(𝑥𝑅𝑦𝑦𝑍)))
5049ss2abdv 3824 . . 3 ((𝑅𝑉𝑍𝑊) → {𝑥 ∣ (∃𝑡 𝑥𝑅𝑡 ∧ ∃𝑡(𝑥𝑅𝑡𝑡𝑍))} ⊆ {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)})
51 suppvalbr 7450 . . 3 ((𝑅𝑉𝑍𝑊) → (𝑅 supp 𝑍) = {𝑥 ∣ (∃𝑡 𝑥𝑅𝑡 ∧ ∃𝑡(𝑥𝑅𝑡𝑡𝑍))})
52 cnvimadfsn 7455 . . . 4 (𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)}
5352a1i 11 . . 3 ((𝑅𝑉𝑍𝑊) → (𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦𝑦𝑍)})
5450, 51, 533sstr4d 3797 . 2 ((𝑅𝑉𝑍𝑊) → (𝑅 supp 𝑍) ⊆ (𝑅 “ (V ∖ {𝑍})))
55 suppimacnvss 7456 . 2 ((𝑅𝑉𝑍𝑊) → (𝑅 “ (V ∖ {𝑍})) ⊆ (𝑅 supp 𝑍))
5654, 55eqssd 3769 1 ((𝑅𝑉𝑍𝑊) → (𝑅 supp 𝑍) = (𝑅 “ (V ∖ {𝑍})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382   = wceq 1631  wex 1852  wcel 2145  {cab 2757  wne 2943  Vcvv 3351  cdif 3720  {csn 4316   class class class wbr 4786  ccnv 5248  cima 5252  (class class class)co 6793   supp csupp 7446
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-supp 7447
This theorem is referenced by:  frnsuppeq  7458  suppun  7466  mptsuppdifd  7468  supp0cosupp0  7486  imacosupp  7487  fdmfisuppfi  8440  fsuppun  8450  fsuppco  8463  gsumval3a  18511  gsumzf1o  18520  gsumzaddlem  18528  gsumzmhm  18544  gsumzoppg  18551  deg1val  24076  suppss3  29842  ffsrn  29844  fpwrelmapffslem  29847  sitgclg  30744  eulerpartlemmf  30777  eulerpartlemgf  30781  fidmfisupp  39909
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