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Theorem rnmptbdd 44521
Description: Boundness of the range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
rnmptbdd.x 𝑥𝜑
rnmptbdd.b (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)
Assertion
Ref Expression
rnmptbdd (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
Distinct variable groups:   𝑦,𝐴,𝑧   𝑦,𝐵,𝑧   𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem rnmptbdd
Dummy variables 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnmptbdd.x . . 3 𝑥𝜑
2 rnmptbdd.b . . . 4 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)
3 breq2 5145 . . . . . 6 (𝑦 = 𝑣 → (𝐵𝑦𝐵𝑣))
43ralbidv 3171 . . . . 5 (𝑦 = 𝑣 → (∀𝑥𝐴 𝐵𝑦 ↔ ∀𝑥𝐴 𝐵𝑣))
54cbvrexvw 3229 . . . 4 (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦 ↔ ∃𝑣 ∈ ℝ ∀𝑥𝐴 𝐵𝑣)
62, 5sylib 217 . . 3 (𝜑 → ∃𝑣 ∈ ℝ ∀𝑥𝐴 𝐵𝑣)
71, 6rnmptbddlem 44520 . 2 (𝜑 → ∃𝑣 ∈ ℝ ∀𝑤 ∈ ran (𝑥𝐴𝐵)𝑤𝑣)
8 breq2 5145 . . . . 5 (𝑣 = 𝑦 → (𝑤𝑣𝑤𝑦))
98ralbidv 3171 . . . 4 (𝑣 = 𝑦 → (∀𝑤 ∈ ran (𝑥𝐴𝐵)𝑤𝑣 ↔ ∀𝑤 ∈ ran (𝑥𝐴𝐵)𝑤𝑦))
10 breq1 5144 . . . . 5 (𝑤 = 𝑧 → (𝑤𝑦𝑧𝑦))
1110cbvralvw 3228 . . . 4 (∀𝑤 ∈ ran (𝑥𝐴𝐵)𝑤𝑦 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
129, 11bitrdi 287 . . 3 (𝑣 = 𝑦 → (∀𝑤 ∈ ran (𝑥𝐴𝐵)𝑤𝑣 ↔ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
1312cbvrexvw 3229 . 2 (∃𝑣 ∈ ℝ ∀𝑤 ∈ ran (𝑥𝐴𝐵)𝑤𝑣 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
147, 13sylib 217 1 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wnf 1777  wral 3055  wrex 3064   class class class wbr 5141  cmpt 5224  ran crn 5670  cr 11111  cle 11253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-mpt 5225  df-cnv 5677  df-dm 5679  df-rn 5680
This theorem is referenced by:  suprclrnmpt  44527  suprubrnmpt2  44528  suprubrnmpt  44529  rnmptbdlem  44531  supxrrernmpt  44703  suprleubrnmpt  44704  supminfrnmpt  44727
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