Mathbox for Scott Fenton < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eqfunresadj Structured version   Visualization version   GIF version

 Description: Law for adjoining an element to restrictions of functions. (Contributed by Scott Fenton, 6-Dec-2021.)
Assertion
Ref Expression
eqfunresadj (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → (𝐹 ↾ (𝑋 ∪ {𝑌})) = (𝐺 ↾ (𝑋 ∪ {𝑌})))

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relres 5851 . 2 Rel (𝐹 ↾ (𝑋 ∪ {𝑌}))
2 relres 5851 . 2 Rel (𝐺 ↾ (𝑋 ∪ {𝑌}))
3 breq 5035 . . . . 5 ((𝐹𝑋) = (𝐺𝑋) → (𝑥(𝐹𝑋)𝑦𝑥(𝐺𝑋)𝑦))
433ad2ant2 1131 . . . 4 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → (𝑥(𝐹𝑋)𝑦𝑥(𝐺𝑋)𝑦))
5 velsn 4544 . . . . . . 7 (𝑥 ∈ {𝑌} ↔ 𝑥 = 𝑌)
6 simp33 1208 . . . . . . . . . 10 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → (𝐹𝑌) = (𝐺𝑌))
76eqeq1d 2803 . . . . . . . . 9 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → ((𝐹𝑌) = 𝑦 ↔ (𝐺𝑌) = 𝑦))
8 simp1l 1194 . . . . . . . . . 10 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → Fun 𝐹)
9 simp31 1206 . . . . . . . . . 10 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → 𝑌 ∈ dom 𝐹)
10 funbrfvb 6699 . . . . . . . . . 10 ((Fun 𝐹𝑌 ∈ dom 𝐹) → ((𝐹𝑌) = 𝑦𝑌𝐹𝑦))
118, 9, 10syl2anc 587 . . . . . . . . 9 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → ((𝐹𝑌) = 𝑦𝑌𝐹𝑦))
12 simp1r 1195 . . . . . . . . . 10 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → Fun 𝐺)
13 simp32 1207 . . . . . . . . . 10 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → 𝑌 ∈ dom 𝐺)
14 funbrfvb 6699 . . . . . . . . . 10 ((Fun 𝐺𝑌 ∈ dom 𝐺) → ((𝐺𝑌) = 𝑦𝑌𝐺𝑦))
1512, 13, 14syl2anc 587 . . . . . . . . 9 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → ((𝐺𝑌) = 𝑦𝑌𝐺𝑦))
167, 11, 153bitr3d 312 . . . . . . . 8 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → (𝑌𝐹𝑦𝑌𝐺𝑦))
17 breq1 5036 . . . . . . . . 9 (𝑥 = 𝑌 → (𝑥𝐹𝑦𝑌𝐹𝑦))
18 breq1 5036 . . . . . . . . 9 (𝑥 = 𝑌 → (𝑥𝐺𝑦𝑌𝐺𝑦))
1917, 18bibi12d 349 . . . . . . . 8 (𝑥 = 𝑌 → ((𝑥𝐹𝑦𝑥𝐺𝑦) ↔ (𝑌𝐹𝑦𝑌𝐺𝑦)))
2016, 19syl5ibrcom 250 . . . . . . 7 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → (𝑥 = 𝑌 → (𝑥𝐹𝑦𝑥𝐺𝑦)))
215, 20syl5bi 245 . . . . . 6 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → (𝑥 ∈ {𝑌} → (𝑥𝐹𝑦𝑥𝐺𝑦)))
2221pm5.32d 580 . . . . 5 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → ((𝑥 ∈ {𝑌} ∧ 𝑥𝐹𝑦) ↔ (𝑥 ∈ {𝑌} ∧ 𝑥𝐺𝑦)))
23 vex 3447 . . . . . 6 𝑦 ∈ V
2423brresi 5831 . . . . 5 (𝑥(𝐹 ↾ {𝑌})𝑦 ↔ (𝑥 ∈ {𝑌} ∧ 𝑥𝐹𝑦))
2523brresi 5831 . . . . 5 (𝑥(𝐺 ↾ {𝑌})𝑦 ↔ (𝑥 ∈ {𝑌} ∧ 𝑥𝐺𝑦))
2622, 24, 253bitr4g 317 . . . 4 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → (𝑥(𝐹 ↾ {𝑌})𝑦𝑥(𝐺 ↾ {𝑌})𝑦))
274, 26orbi12d 916 . . 3 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → ((𝑥(𝐹𝑋)𝑦𝑥(𝐹 ↾ {𝑌})𝑦) ↔ (𝑥(𝐺𝑋)𝑦𝑥(𝐺 ↾ {𝑌})𝑦)))
28 resundi 5836 . . . . 5 (𝐹 ↾ (𝑋 ∪ {𝑌})) = ((𝐹𝑋) ∪ (𝐹 ↾ {𝑌}))
2928breqi 5039 . . . 4 (𝑥(𝐹 ↾ (𝑋 ∪ {𝑌}))𝑦𝑥((𝐹𝑋) ∪ (𝐹 ↾ {𝑌}))𝑦)
30 brun 5084 . . . 4 (𝑥((𝐹𝑋) ∪ (𝐹 ↾ {𝑌}))𝑦 ↔ (𝑥(𝐹𝑋)𝑦𝑥(𝐹 ↾ {𝑌})𝑦))
3129, 30bitri 278 . . 3 (𝑥(𝐹 ↾ (𝑋 ∪ {𝑌}))𝑦 ↔ (𝑥(𝐹𝑋)𝑦𝑥(𝐹 ↾ {𝑌})𝑦))
32 resundi 5836 . . . . 5 (𝐺 ↾ (𝑋 ∪ {𝑌})) = ((𝐺𝑋) ∪ (𝐺 ↾ {𝑌}))
3332breqi 5039 . . . 4 (𝑥(𝐺 ↾ (𝑋 ∪ {𝑌}))𝑦𝑥((𝐺𝑋) ∪ (𝐺 ↾ {𝑌}))𝑦)
34 brun 5084 . . . 4 (𝑥((𝐺𝑋) ∪ (𝐺 ↾ {𝑌}))𝑦 ↔ (𝑥(𝐺𝑋)𝑦𝑥(𝐺 ↾ {𝑌})𝑦))
3533, 34bitri 278 . . 3 (𝑥(𝐺 ↾ (𝑋 ∪ {𝑌}))𝑦 ↔ (𝑥(𝐺𝑋)𝑦𝑥(𝐺 ↾ {𝑌})𝑦))
3627, 31, 353bitr4g 317 . 2 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → (𝑥(𝐹 ↾ (𝑋 ∪ {𝑌}))𝑦𝑥(𝐺 ↾ (𝑋 ∪ {𝑌}))𝑦))
371, 2, 36eqbrrdiv 5635 1 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → (𝐹 ↾ (𝑋 ∪ {𝑌})) = (𝐺 ↾ (𝑋 ∪ {𝑌})))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112   ∪ cun 3882  {csn 4528   class class class wbr 5033  dom cdm 5523   ↾ cres 5525  Fun wfun 6322  ‘cfv 6328 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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-res 5535  df-iota 6287  df-fun 6330  df-fn 6331  df-fv 6336 This theorem is referenced by:  eqfunressuc  33119
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