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Theorem ssrnres 5067
Description: Subset of the range of a restriction. (Contributed by NM, 16-Jan-2006.)
Assertion
Ref Expression
ssrnres (𝐵 ⊆ ran (𝐶𝐴) ↔ ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵)

Proof of Theorem ssrnres
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss2 3356 . . . . 5 (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)
2 rnss 4853 . . . . 5 ((𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵) → ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐴 × 𝐵))
31, 2ax-mp 5 . . . 4 ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐴 × 𝐵)
4 rnxpss 5056 . . . 4 ran (𝐴 × 𝐵) ⊆ 𝐵
53, 4sstri 3164 . . 3 ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ 𝐵
6 eqss 3170 . . 3 (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ (ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ 𝐵𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵))))
75, 6mpbiran 940 . 2 (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)))
8 ssid 3175 . . . . . . . 8 𝐴𝐴
9 ssv 3177 . . . . . . . 8 𝐵 ⊆ V
10 xpss12 4730 . . . . . . . 8 ((𝐴𝐴𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (𝐴 × V))
118, 9, 10mp2an 426 . . . . . . 7 (𝐴 × 𝐵) ⊆ (𝐴 × V)
12 sslin 3361 . . . . . . 7 ((𝐴 × 𝐵) ⊆ (𝐴 × V) → (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐶 ∩ (𝐴 × V)))
1311, 12ax-mp 5 . . . . . 6 (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐶 ∩ (𝐴 × V))
14 df-res 4635 . . . . . 6 (𝐶𝐴) = (𝐶 ∩ (𝐴 × V))
1513, 14sseqtrri 3190 . . . . 5 (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐶𝐴)
16 rnss 4853 . . . . 5 ((𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐶𝐴) → ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐶𝐴))
1715, 16ax-mp 5 . . . 4 ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐶𝐴)
18 sstr 3163 . . . 4 ((𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) ∧ ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐶𝐴)) → 𝐵 ⊆ ran (𝐶𝐴))
1917, 18mpan2 425 . . 3 (𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) → 𝐵 ⊆ ran (𝐶𝐴))
20 ssel 3149 . . . . . . 7 (𝐵 ⊆ ran (𝐶𝐴) → (𝑦𝐵𝑦 ∈ ran (𝐶𝐴)))
21 vex 2740 . . . . . . . 8 𝑦 ∈ V
2221elrn2 4865 . . . . . . 7 (𝑦 ∈ ran (𝐶𝐴) ↔ ∃𝑥𝑥, 𝑦⟩ ∈ (𝐶𝐴))
2320, 22syl6ib 161 . . . . . 6 (𝐵 ⊆ ran (𝐶𝐴) → (𝑦𝐵 → ∃𝑥𝑥, 𝑦⟩ ∈ (𝐶𝐴)))
2423ancrd 326 . . . . 5 (𝐵 ⊆ ran (𝐶𝐴) → (𝑦𝐵 → (∃𝑥𝑥, 𝑦⟩ ∈ (𝐶𝐴) ∧ 𝑦𝐵)))
2521elrn2 4865 . . . . . 6 (𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)))
26 elin 3318 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
27 opelxp 4653 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
2827anbi2i 457 . . . . . . . 8 ((⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ (𝑥𝐴𝑦𝐵)))
2921opelres 4908 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶𝑥𝐴))
3029anbi1i 458 . . . . . . . . 9 ((⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴) ∧ 𝑦𝐵) ↔ ((⟨𝑥, 𝑦⟩ ∈ 𝐶𝑥𝐴) ∧ 𝑦𝐵))
31 anass 401 . . . . . . . . 9 (((⟨𝑥, 𝑦⟩ ∈ 𝐶𝑥𝐴) ∧ 𝑦𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ (𝑥𝐴𝑦𝐵)))
3230, 31bitr2i 185 . . . . . . . 8 ((⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ (𝑥𝐴𝑦𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴) ∧ 𝑦𝐵))
3326, 28, 323bitri 206 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴) ∧ 𝑦𝐵))
3433exbii 1605 . . . . . 6 (∃𝑥𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥(⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴) ∧ 𝑦𝐵))
35 19.41v 1902 . . . . . 6 (∃𝑥(⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴) ∧ 𝑦𝐵) ↔ (∃𝑥𝑥, 𝑦⟩ ∈ (𝐶𝐴) ∧ 𝑦𝐵))
3625, 34, 353bitri 206 . . . . 5 (𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ (∃𝑥𝑥, 𝑦⟩ ∈ (𝐶𝐴) ∧ 𝑦𝐵))
3724, 36syl6ibr 162 . . . 4 (𝐵 ⊆ ran (𝐶𝐴) → (𝑦𝐵𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵))))
3837ssrdv 3161 . . 3 (𝐵 ⊆ ran (𝐶𝐴) → 𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)))
3919, 38impbii 126 . 2 (𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ 𝐵 ⊆ ran (𝐶𝐴))
407, 39bitr2i 185 1 (𝐵 ⊆ ran (𝐶𝐴) ↔ ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵)
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105   = wceq 1353  wex 1492  wcel 2148  Vcvv 2737  cin 3128  wss 3129  cop 3594   × cxp 4621  ran crn 4624  cres 4625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-xp 4629  df-rel 4630  df-cnv 4631  df-dm 4633  df-rn 4634  df-res 4635
This theorem is referenced by:  rninxp  5068
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