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Theorem ssrnres 4788
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 3183 . . . . 5 (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)
2 rnss 4589 . . . . 5 ((𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵) → ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐴 × 𝐵))
31, 2ax-mp 7 . . . 4 ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐴 × 𝐵)
4 rnxpss 4779 . . . 4 ran (𝐴 × 𝐵) ⊆ 𝐵
53, 4sstri 2979 . . 3 ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ 𝐵
6 eqss 2985 . . 3 (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ (ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ 𝐵𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵))))
75, 6mpbiran 856 . 2 (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)))
8 ssid 2989 . . . . . . . 8 𝐴𝐴
9 ssv 2990 . . . . . . . 8 𝐵 ⊆ V
10 xpss12 4470 . . . . . . . 8 ((𝐴𝐴𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (𝐴 × V))
118, 9, 10mp2an 410 . . . . . . 7 (𝐴 × 𝐵) ⊆ (𝐴 × V)
12 sslin 3188 . . . . . . 7 ((𝐴 × 𝐵) ⊆ (𝐴 × V) → (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐶 ∩ (𝐴 × V)))
1311, 12ax-mp 7 . . . . . 6 (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐶 ∩ (𝐴 × V))
14 df-res 4382 . . . . . 6 (𝐶𝐴) = (𝐶 ∩ (𝐴 × V))
1513, 14sseqtr4i 3003 . . . . 5 (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐶𝐴)
16 rnss 4589 . . . . 5 ((𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐶𝐴) → ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐶𝐴))
1715, 16ax-mp 7 . . . 4 ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐶𝐴)
18 sstr 2978 . . . 4 ((𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) ∧ ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐶𝐴)) → 𝐵 ⊆ ran (𝐶𝐴))
1917, 18mpan2 409 . . 3 (𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) → 𝐵 ⊆ ran (𝐶𝐴))
20 ssel 2964 . . . . . . 7 (𝐵 ⊆ ran (𝐶𝐴) → (𝑦𝐵𝑦 ∈ ran (𝐶𝐴)))
21 vex 2575 . . . . . . . 8 𝑦 ∈ V
2221elrn2 4601 . . . . . . 7 (𝑦 ∈ ran (𝐶𝐴) ↔ ∃𝑥𝑥, 𝑦⟩ ∈ (𝐶𝐴))
2320, 22syl6ib 154 . . . . . 6 (𝐵 ⊆ ran (𝐶𝐴) → (𝑦𝐵 → ∃𝑥𝑥, 𝑦⟩ ∈ (𝐶𝐴)))
2423ancrd 313 . . . . 5 (𝐵 ⊆ ran (𝐶𝐴) → (𝑦𝐵 → (∃𝑥𝑥, 𝑦⟩ ∈ (𝐶𝐴) ∧ 𝑦𝐵)))
2521elrn2 4601 . . . . . 6 (𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)))
26 elin 3151 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
27 opelxp 4399 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
2827anbi2i 438 . . . . . . . 8 ((⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ (𝑥𝐴𝑦𝐵)))
2921opelres 4642 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶𝑥𝐴))
3029anbi1i 439 . . . . . . . . 9 ((⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴) ∧ 𝑦𝐵) ↔ ((⟨𝑥, 𝑦⟩ ∈ 𝐶𝑥𝐴) ∧ 𝑦𝐵))
31 anass 387 . . . . . . . . 9 (((⟨𝑥, 𝑦⟩ ∈ 𝐶𝑥𝐴) ∧ 𝑦𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ (𝑥𝐴𝑦𝐵)))
3230, 31bitr2i 178 . . . . . . . 8 ((⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ (𝑥𝐴𝑦𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴) ∧ 𝑦𝐵))
3326, 28, 323bitri 199 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴) ∧ 𝑦𝐵))
3433exbii 1510 . . . . . 6 (∃𝑥𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥(⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴) ∧ 𝑦𝐵))
35 19.41v 1796 . . . . . 6 (∃𝑥(⟨𝑥, 𝑦⟩ ∈ (𝐶𝐴) ∧ 𝑦𝐵) ↔ (∃𝑥𝑥, 𝑦⟩ ∈ (𝐶𝐴) ∧ 𝑦𝐵))
3625, 34, 353bitri 199 . . . . 5 (𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ (∃𝑥𝑥, 𝑦⟩ ∈ (𝐶𝐴) ∧ 𝑦𝐵))
3724, 36syl6ibr 155 . . . 4 (𝐵 ⊆ ran (𝐶𝐴) → (𝑦𝐵𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵))))
3837ssrdv 2976 . . 3 (𝐵 ⊆ ran (𝐶𝐴) → 𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)))
3919, 38impbii 121 . 2 (𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ 𝐵 ⊆ ran (𝐶𝐴))
407, 39bitr2i 178 1 (𝐵 ⊆ ran (𝐶𝐴) ↔ ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵)
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102   = wceq 1257  wex 1395  wcel 1407  Vcvv 2572  cin 2941  wss 2942  cop 3403   × cxp 4368  ran crn 4371  cres 4372
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-pow 3952  ax-pr 3969
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-v 2574  df-un 2947  df-in 2949  df-ss 2956  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3790  df-opab 3844  df-xp 4376  df-rel 4377  df-cnv 4378  df-dm 4380  df-rn 4381  df-res 4382
This theorem is referenced by:  rninxp  4789
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